WORKS   OF 
FREDERICK  H.  GETMAN,  Ph.D. 


PUBLISHED    BY 


JOHN  WILEY  &   SONS,  Inc. 


Outlines  of  Theoretical  Chemistry. 

Second  Edition,  Thoroughly  Revised,    xv  +  539 
pages,  5i  X  8.    Ill  figures.    Cloth,  $3.50  net. 

An  Introduction  to  Physical  Science. 

vi  +  257  pages,   5  X  7*.     129  figures.     Cloth, 
$1.50  net. 

Laboratory  Exercises  in  Physical  Chemistry. 

Second    Edition,     Revised.       x  +  285     pages, 
5  X  7J.     115  figures.    Cloth,  $2.00  net. 


OUTLINES 


OF 


THEORETICAL  CHEMISTRY 


BY 

FREDERICK  H.   GETMAN,   Ph.D.  (Johns  Hopkins) 

Formerly  Associate  Professor  of  Chemistry  in  Bryn  Mawr  College 


SECOND  EDITION,  THOROUGHLY 
REVISED  AND  ENLARGED 


NEW  YORK 

JOHN  WILEY  &  SONS,  INC. 

LONDON:   CHAPMAN   &   HALL,  LIMITED 
1918 


COPYRIGHT,  1913,  1918, 

BY 
FREDERICK  H.  GETMAN 


Stanhope  jpress 

F.    H.GILSON   COMPANY 
BOSTON.  U.S.A. 


To 


WHOSE   TENDER   SYMPATHY   HAS  HELPED  TO 

LIGHTEN  THE  DARKNESS  OF  THE  DAYS 

DURING  WHICH  THESE  PAGES 

WERE  WRITTEN 


385126 


"I  look  upon  the  common  operations  and  practices  of  chymists,  almost 
as  I  do  on  the  letters  of  the  alphabet,  without  whose  knowledge  'tis  very 
hard  for  a  man  to  become  a  philosopher;  and  yet  that  knowledge  is  very  far 
from  being  sufficient  to  make  him  one." 

ROBERT  BOYLE.     (The  Sceptical  Chymist.) 


PREFACE   TO  SECOND  EDITION. 


"  The  theoretical  side  of  physical  chemistry  is  and  will  probably  remain  the 
dominant  one;  it  is  by  this  peculiarity  that  it  has  exerted  such  a  great  in- 
fluence upon  the  neighboring  sciences,  pure  and  applied,  and  on  this  ground 
physical  chemistry  may  be  regarded  as  an  excellent  school  of  exact  reasoning 
for  all  students  of  natural  sciences."  —  Arrhenius. 

The  demand  for  a  second  edition  of  this  book  has  not  only 
afforded  the  author  an  opportunity  to  thoroughly  revise  the 
original  text,  but  also  has  made  it  possible  to  include  such  new 
material  as  should  properly  find  a  place  in  an  introductory  text- 
book ot  theoretical  chemistry. 

The  arduousness  of  the  task  of  revision  and  amplification  has 
been  appreciably  lightened  by  the  helpful  criticisms  and  valuable 
suggestions  which  have  been  received  from  those  who  have  used 
the  first  edition  with  their  classes. 

So  numerous  were  the  additional  topics  suggested  that  the 
author  found  himself  confronted  with  a  veritable  embarrassment 
of  riches,  and  not  the  least  difficult  part  of  his  task  has  been  the 
attempt  to  weave  in  as  many  of  these  suggestions  as  seemed  to  be 
consistent  with  a  well-balanced  presentation  of  the  entire  subject. 

The  features  which  distinguish  this  edition  from  the  preceding 
edition  may  be  briefly  summarized  as  follows :  — 

1.  The  necessity  of  introducing  a  short  chapter  on  the  mod- 
ern conception  of  the  atom  and  its  structure  involved  the 
further  necessity  of  including  a  preliminary  chapter  treat- 
ing of  those  radioactive  phenomena  upon  which  the  greater 
part  of  our  present  atomic  theory  is  based. 

2.  The   chapter  on   solids   has   been   practically  rewritten: 
the  space  formerly  devoted  to  an  outline  of  crystallog- 

vii 


Vlll  PREFACE 

raphy  being  devoted  in  the  present  edition  to  a  discussion 
of  the  absorption  of  heat  by  crystalline  solids  and  the  bear- 
ing of  X-ray  spectra  on  crystalline  form. 

3.  The  increasing  importance  of  colloidal  phenomena,  not 
only  to  the  chemist  but  also  to  the  biologist,  to  the  physician, 
and  to  the  technologist,  has  made  it  seem  desirable  to  rewrite 
the  entire  chapter  devoted  to  the  chemistry  of  colloids. 

4.  The  Brownian  movement  and  its  bearing  upon  the  exist- 
ence of  molecules  has  been  briefly  presented  in  a  separate 
chapter  in  order  to  emphasize  the  importance  of  the  bril- 
liant experimental  work  of  Perrin  and  others  in  confirming 
the  kinetic  theory. 

5.  The   chapter  treating  of  electromotive   force   has   been 
enlarged  so  as  to  include  a  discussion  of  some  of  the  more 
valuable  methods  which  have  been  proposed  for  deter- 
mining junction  potentials  and  also  to  point  out  several 
useful  applications  of  concentration  cells. 

6.  An  entirely  new  chapter  in  which  an  attempt  has  been  made 
to  present  the  salient  facts  and  more  important  theories 
of   photochemistry  in  succinct  form  replaces  the  former 
chapter  treating  of   the   relations   between   radiant  and 
chemical  energy. 

Among  the  books  to  which  the  author  has  had  frequent  re- 
course in  the  preparation  of  this  edition  should  be  mentioned 
Rutherford's  " Radioactive  Substances  and  their  Radiations,"  W. 
H.  and  W.  L.  Bragg's  "X-Rays  and  Crystal  Structure,"  Freund- 
lich's  "Kapillarchemie,"  Perrin's  "Les  Atomes,"  and  Sheppard's 
"Photochemistry." 

It  is  with  a  deep  sense  of  gratitude  that  acknowledgment  is 
made  to  all  of  those  friends  who  have  offered  criticisms  of  the  old 
edition  and  suggestions  for  the  new.  Special  thanks  are  due 
to  Dr.  W.  D.  Harkins  of 'the  University  of  Chicago  not  only  for 
suggestions  but  for  permission  to  quote  extensively  from  his 
papers,  to  Dr.  J.  Howard  Mathews  of  the  University  of  Wis- 
consin for  numerous  helpful  suggestions,  to  Dr.  Walter  A.  Patrick 
of  Johns  Hopkins  University  for  criticisms  of  the  former  chapter 


PREFACE  IX 

on  colloids  and  to  Mr.  John  McGavack  for  his  conscientious  work 
in  checking  the  answers  to  all  of  the  problems.  In  the  preparation 
of  the  indices  of  names  and  subjects  the  author  is  indebted  to  his 
wife  and  to  Miss  Mary  K.  Pease  who  have  given  valuable  assist- 
ance in  that  wearisome  and  exacting  task.  To  the  publishers, 
Messrs.  John  Wiley  and  Sons,  Inc.,  acknowledgment  is  made 
of  their  kindness  in  permitting  the  use  of  Fig.  68  taken  from 
Chamot's  "Elementary  Chemical  Microscopy." 

FREDERICK  H.  GETMAN.; 
STAMFORD,  CONN. 
Aug.  7,  1918. 


PREFACE. 


"  The  last  thing  that  we  find  in  making  a  book  is  to  know  what  we  must 
put  first."  —  PASCAL. 

THE  present  book  is  designed  to  meet  the  requirements  of 
classes  beginning  the  study  of  theoretical  or  physical  chemistry. 
A  working  knowledge  of  elementary  chemistry  and  physics  has 
been  presupposed  in  the  presentation  of  the  subject,  the  introduc- 
tory chapter  being  the  only  portion  of  the  book  in  which  space  is 
devoted  to  a  review  of  principles  with  which  the  student  is  assumed 
to  be  already  fairly  familiar.  With  the  exception  of  a  few  para- 
graphs in  which  the  application  of  the  calculus  is  unavoidable, 
no  use  is  made  of  the  higher  mathematics,  so  that  the  book  should 
be  intelligible  to  the  student  of  very  moderate  mathematical 
attainments.  Wherever  the  calculus  has  been  employed,  the 
student  who  is  unfamiliar  with  this  useful  tool  must  accept  the 
correctness  of  the  results  without  attempting  to  follow  the  suc- 
cessive operations  by  which  they  are  obtained. 

The  contributions  to  our  knowledge  in  the  domain  of  physical 
chemistry  have  increased  with  such  rapidity  within  recent  years, 
that  the  prospective  author  of  a  general  text  book  finds  himself 
confronted  with  the  vexing  problem  of  what  to  omit  rather  than 
what  to  include.  In  selecting  material  for  this  book,  the  author 
has  been  guided  in  large  measure  by  his  own  experience  in  teaching 
theoretical  chemistry  to  beginners  and  to  advanced  students. 
The  attempt  has  been  made  to  present  the  more  difficult  portions 
of  the  subject,  such  as  the  osmotic  theory  of  solutions,  the  laws 
of  equilibrium  and  chemical  action,  and  the  principles  of  electro- 
chemistry, in  a  clear  and  logical  manner.  While  the  treatment 
of  each  topic  is  necessarily  brief  yet  the  effort  has  been  made  to 
avoid  the  sacrifice  of  clearness  to  brevity. 


CONTENTS 


CHAP.  PAGE 

PREFACE  TO  SECOND  EDITION vii 

PREFACE xi 

I.   Fundamental  Principles 1 

II.   Classification  of  the  Elements 20 

III.  The  Electron  Theory 31 

IV.  Radioactivity 42 

V.   Atomic  Structure 57 

VI.   Gases ; 72 

VII.  Liquids *. 104 

VIII.   Solids 153 

IX.   Solutions 167 

X.   Dilute  Solutions  and  Osmotic  Pressure 187 

XI.  Association,  Dissociation  and  Solvation 225 

XII.   Colloids 237 

XIII.  Molecular  Reality 279 

XIV.  Thermochemistry 286 

XV.   Homogeneous  Equilibrium 312 

XVI.   Heterogeneous  Equilibrium 328 

XVII.  Chemical  Kinetics 359 

XVIII.   Electrical  Conductance 385 

XIX.   Electrolytic  Equilibrium  and  Hydrolysis 422 

XX.   Electromotive  Force 446 

XXI.   Electrolysis  and  Polarization 492 

XXII.   Photochemistry 504 

INDEX  OF  NAMES 529 

INDEX  OF  SUBJECTS  .  .  533 


THEORETICAL  CHEMISTRY. 


CHAPTER  I. 
FUNDAMENTAL   PRINCIPLES. 

Theoretical  Chemistry.  That  portion  of  the  science  of  chem- 
istry which  has  for  its  object  the  study  of  the  laws  controlling 
chemical  phenomena  is  called  theoretical  or  physical  chemistry. 
The  first  attempt  to  summarize  the  more  important  facts  and 
ideas  underlying  the  science  of  chemistry  was  made  by  Dalton  in 
1808  in  his  "New  System  of  Chemical  Philosophy."  The  birth 
of  the  science  of  theoretical  chemistry  may  be  considered  to  be 
coeval  with  the  appearance  of  Dalton's  epoch-making  book. 

Theoretical  chemistry  is  concerned  with  the  great  generaliza- 
tions of  chemical  science  and  bears  the  same  relation  to  chemistry 
that  philosophy  bears  to  the  whole  body  of  scientific  truth;  it 
aims  to  systematize  all  of  the  established  facts  of  chemistry 
and  to  discover  the  laws  governing  the  various  phenomena  of 
chemical  action. 

Law,  Hypothesis  and  Theory.  The  science  of  chemistry  is 
based  upon  experimentally  established  facts.  When  a  number 
of  facts  have  been  collected  and  classified  we  may  proceed  to  draw 
inferences  as  to  the  behavior  of  systems  under  conditions  which 
have  not  been  investigated.  This  process  of  reasoning  by  analogy 
we  term  generalization  and  the  conclusion  .reached  we  call  a  law. 
It  is  apparent  that  a  law  is  not  an  expression  of  an  infallible  truth, 
but  it  is  rather  a  condensed  statement  of  facts  which  have  been 
discovered  by  experiment.  It  enables  us  to  predict  results  with- 
out recourse  to  experiment.  The  fewer  the  number  of  cases  in 
which  a  law  has  been  found  to  be  invalid,  the  greater  becomes  our 
confidence  in  it,  until  eventually  it  may  come  to  be  regarded  as 
tantamount  to  a  statement  of  fact. 

1 


2  THEORETICAL  CHEMISTRY 

Natural  laws  may  be  discovered  by  the  correlation  of  experi- 
mentally determined  facts,  as  outlined  above,  or  by  means  of  a 
speculation  as  to  the  probable  cause 'of  the  phenomena  in  question. 
Such  a  speculation  in  regard  to  the  cause  of  a  phenomenon  is 
called  an  hypothesis. 

After  an  hypothesis  has  been  subjected  to  the  test  of  experiment 
and  has  been  shown  to  apply  to  a  large  number  of  closely  related 
phenomena  it  is  termed  a  theory. 

In  his  address  to  the  British  Association  (Dundee,  1912),  Profes- 
sor Senier  has  this  to  say :  "  While  the  method  of  discovery  in  chem- 
istry may  be  described  generally,  as  inductive,  still  all  the  modes  of 
inference  which  have  come  down  to  us  from  Aristotle,  analogical, 
inductive,  and  deductive,  are  freely  used.  An  hypothesis  is  framed 
which  is  then  tested,  directly  or  indirectly,  by  observation  and 
experiment.  All  the  skill,  and  all  the  resource  the  inquirer  can 
command,  are  brought  into  his  service;  his  work  must  be  accurate; 
and  with  unqualified  devotion  to  truth  he  abides  by  the  result, 
and  the  hypothesis  is  established,  and  becomes  a  part  of  the 
theory  of  science,  or  is  rejected  or  modified. " 

Elements  and  Compounds.  All  definite  chemical  substances 
are  divided  into  two  classes,  elements  and  compounds. 

Robert  Boyle  was  the  first  to.  make  this  distinction.  He  de- 
fined an  element  as  a  substance  which  is  incapable  of  resolution 
into  anything  simpler.  The  substances  formed  by  the  chemical 
combination  of  two  or  more  elements  he  termed  chemical  com- 
pounds. This  definition  of  an  element  as  given  by  Boyle  was 
later  proposed  by  Lavoisier  and,  notwithstanding  the  vast  accumu- 
lation of  scientific  knowledge  since  their  time,  the  definition  re- 
mains very  satisfactory  today. 

At  the  present ]time  we  have  a  group  of  about  eighty  substances 
which  have  resisted  all  efforts  to  decompose  them  into  simpler 
substances.  These  are  the  so-called  chemical  elements.  It 
should  be  borne  in  mind,  however,  that  because  we  have  failed  to 
resolve  these  substances  into  simpler  forms  of  matter,  we  are  not 
warranted  in  maintaining  that  such  resolution  may  not  be  effected 
in  the  future. 

Recent  investigations  of  the  radioactive  elements  have  shown 


FUNDAMENTAL  PRINCIPLES  3 

that  they  are  continuously  undergoing  a  series  of  transformations, 
one  of  the  products  of  which  is  the  inactive  element  helium.  This 
behavior  is  contrary  to  the  old  view  that  transformation  of  one 
element  into  another  is  impossible.  At  first  the  attempt  was 
made  to  explain  it  by  assuming  that  the  radioactive  element  was 
a  compound  of  helium  with  another  element,  but  since  the  radio- 
active elements  possess  all  of  the  properties  characteristic  of 
elements  as  distinguished  from  compounds,  and  find  appropriate 
places  in  the  periodic  table  of  Mendeleeff,  the  "compound  theory" 
must  be  abandoned.  Uranium  and  thorium,  the  heaviest  ele- 
ments known,  appear  to  be  undergoing  a  process  of  spontaneous 
disintegration  over  which  we  have  no  control.  The  products 
of  this  disintegration  have  filled  the  gap  in  the  periodic  table 
between  thorium  and  lead  with  about  thirty  new  elements,  each 
of  which  is  in  turn  undergoing  transformations  similar  to  those 
of  the  parent  elements.  Professor  Soddy  *  says:  "In  spite  of 
the  existence  at  one  time  of  a  vague  belief  (  a  belief  which  has  no 
foundation),  that  all  matter  may  be  to  a  certain  extent  radioactive, 
just  as  all  matter  is  believed  to  be  to  a  certain  extent  magnetic,  it 
is  recognized  today  that  radioactivity  is  an  exceedingly  rare  prop- 
erty of  matter." 

Notwithstanding  these  remarkable  discoveries,  we  may  still  hold 
to  the  idea  of  an  element  as  suggested  by  Boyle  and  Lavoisier. 
Professor  Walker  says:  "The  elements  form  a  group  of  substances, 
singular  not  only  with  respect  to  the  resistance  which  they  offer 
to  decomposition,  but  also  with  respect  to  certain  regularities  dis- 
played by  them  and  not  shared  by  substances  which  are  designated 
as  compounds." 

Law  of  the  Conservation  of  Mass.     In  1774,  as  the  result  of 
a  series  of  experiments,  Lavoisier  established  the  law  of  the  con- 
servation of  mass  which  may  be  stated  as  follows :     In  a  chemical 
reaction  the  total  mass  of  the  reacting  substances  is  equal  to  the  total 
mass  of  the  products  of  the  reaction.     It  is  sometimes  stated  thus:  — 
the  total  mass  of  the  universe  is  a  constant;  but  this  form  of  state- 
ment is  open  to  the  objection  that  we  have  no  means  of  verification; 
it  is  a  statement  of  a  fact  which  transcends  our  experience. 
*  Chemistry  of  the  Radio-Elements,  p.  2. 


4  THEORETICAL  CHEMISTRY 

The  law  of  the  conservation  of  mass  has  been  subjected  to  most 
rigid  investigation  by  Landolt  *  in  a  series  of  experiments  extend- 
ing over  a  period  of  fifteen  years. 

The  reacting   substances,  AB   and   CD, 
^  were  placed   in  the  two  arms   of    the  in- 

verted U-tube  shown  in  Fig.  1  which  was 
then  sealed  at  S  and  the  whole  weighed 
upon  an  extremely  sensitive  balance.  The 
vessel  was  then  inverted  when  the  following 
reaction  took  place :  — 

AB  +  CD-+AD  +  CB. 

When  the  reaction  was  complete  and  suf- 
ficient time  had  elapsed  to  allow  the  vessel 
AB|  I  ^zr-nicD  to  return  to  its  original  volume  (this  some- 
times required  nearly  three  weeks),  it  was 
weighed  again  using  every  precaution  to 
avoid  errors  and  any  gain  or  loss  in  weight 
noted. 


Fig.  1. 


Landolt  concluded  from  the  thirty  or  more  reactions  which  he 
studied  that  the  gain  or  loss  in  weight  was  less  than  one  ten- 
millionth  of  the  total  weight,  f 

Law  of  Definite  Proportions.  The  enunciation  of  the  law  of 
the  conservation  of  mass  and  the  introduction  of  the  balance  into 
the  chemical  laboratory  marked  the  beginning  of  a  new  era  in  the 
history  of  chemistry,  —  the  era  of  quantitative  chemistry.  As 
the  result  of  painstaking  experimental  work,  Richter  and  Proust 
announced  the  law  of  definite  proportions  about  the  beginning 
of  the  nineteenth  century.  Tnis  law  may  be  expressed  thus: 
A  definite  chemical  compound  always  contains  the  same  elements 
united  in  the  same  proportion  by  weight. 

Shortly  after  the  enunciation  of  this  law  its  truth  was  questioned 
by  the  French  chemist  Berthollet.|  From  the  results  of  a  series 


*  Zeit.  phys.  chem.,  12,  1  (1893);  55,  589  (1906). 

f  An  excellent  summary  of  this  important  investigation  will  be  found  in 
the  Journal  de  Chimie  physique,  6,  625  (1908).  • 
J  Essai  de  statique  chimique  (1803). 


FUNDAMENTAL  PRINCIPLES  5 

of  brilliant  experiments,  he  became  convinced  that  chemical  reac- 
tions are  largely  controlled  by  the  relative  amounts  of  the  react- 
ing substances.  As  we  shall  see  later,  he  really  foreshadowed  the 
work  of  Guldberg  and  Waage  who  were  the  first  to  correctly  for- 
mulate the  influence  of  mass  on  a  chemical  reaction.  Berthollet 
argued  that  when  two  elements  unite  to  form  a  compound,  the 
proportion  of  one  of  the  elements  in  the  compound  is  conditioned 
solely  by  the  amount  of  that  element  which  is  available.  This 
led  to  the  celebrated  controversy  between  Berthollet  and  Proust 
which  finally  resulted  in  the  establishment  of  the  latter's  original 
statement.  Subsequent  investigation  has  only  strengthened  our 
faith  in  the  law  of  definite  proportions. 

Law  of  Multiple  Proportions.  Elements  are  known  to  unite 
in  more  than  one  proportion  by  weight.  Dalton  analyzed  the 
two  compounds  of  carbon  and  hydrogen,  methane  and  ethylene, 
and  found  that  the  ratio  of  the  weights  of  carbon  to  hydrogen  in 
the  former  was  6 :  2  while  in  the  latter  it  was  6:1.  That  is,  for  the 
same  weight  of  carbon,  the  weights  of  the  hydrogen  in  the  two 
compounds  were  in  the  ratio  2:1. 

A  large  number  of  compounds  were  examined  and  similar 
simple  ratios  between  the  masses  of  the  constituent  elements 
were  found.  As  a  result  of  these  observations,  Dalton  *  formu- 
lated in  1808  the  law  of  multiple  proportions,  as  follows:  When 
two  elements  unite  in  more  than  one  proportion,  for  a  fixed  mass  of 
one  element  the  masses  of  the  other  element  bear  to  each  other  a  simple 
ratio.  Notwithstanding  the  fact  that  Dalton  was  a  careless 
experimenter  the  subsequent  investigations  of  Marignac  and 
others  have  established  the  validity  of  his  law. 

Law  of  Combining  Proportions.  Dalton  pointed  out  that  it  is 
possible  to  assign  to  every  element  a  definite  relative  weight  with 
which  it  enters  into  chemical  combination.  He  observed  that 
the  weights  or  simple  multiples  of  the  weights  of  the  different 
elements  which  unite  with  a  given  weight  of  a  definite  element, 
represent  the  weights  of  the  different  elements  which  combine 
with  each  other.  The  weights  of  the  elements  which  combine 
with  each  other  are  termed  their  combining  weights.  This  com- 
*  A  New  System  of  Chemical  Philosophy  (1808). 


6  THEORETICAL  CHEMISTRY 

prehensive  law  of  chemical  combination  may  be  stated  as  follows: 
Elements  combine  in  the  ratio  of  their  combining  weights  or  in 
simple  multiples  of  this  ratio.  It  will  be  observed  that  this  law 
really  includes  the  law  of  definite  and  the  law  of  multiple  pro- 
portions. 

If  we  assume  the  combining  weight  of  hydrogen  to  be  unity, 
the  combining  weights  of  chlorine,  oxygen  and  sulphur  will  be 
35.5,  8  and  16  respectively.  These  numbers  represent  the  ratios 
in  which  the  elements  substitute  each  other  in  chemical  com- 
pounds. Hydrochloric  acid,  for  example,  contains  35.5  parts  by 
weight  of  chlorine  to  1  part  by  weight  of  hydrogen  and  when 
oxygen  is  substituted  for  chlorine,  forming  water,  the  new  com- 
pound contains  8  parts  by  weight  of  oxygen  to  1  part  by  weight 
of  hydrogen.  Similarly,  if  the  oxygen  be  substituted  by  sulphur, 
forming  hydrogen  sulphide,  there  will  be  found  16  parts  by  weight 
of  sulphur  to  1  part  by  weight  of  hydrogen.  We  may  say,  then, 
that  35.5  parts  of  chlorine,  8  parts  of  oxygen  and  16  parts  of  sul- 
phur are  equivalent. 

A  chemical  equivalent  may  be  defined  as  the  weight  of  an  element 
which  is  necessary  to  combine  with  or  displace  1  part  by  weight 
of  hydrogen. 

The  Atomic  Theory.  In  very  early  times  two  different  views 
were  entertained  by  opposing  schools  of  Greek  philosophers  as  to 
the  mechanical  constitution  of  matter.  According  to  the  school 
of  Plato  and  Aristotle,  matter  was  thought  to  be  continuous 
within  the  space  it  appears  to  fill  and  to  be  capable  of  indefinite 
subdivision.  According  to  the  other  school,  first  taught  by 
Leucippus,  and  afterwards  by  Democritus  and  Epicurus,  matter 
was  considered  to  be  made  up  of  primordial,  extremely  minute 
particles,  distinct  and  separable  from  each  other  but  in  themselves 
incapable  of  division.  These  ultimate  particles  were  called  atoms 
(iJaTo/uios),  signifying  something  indivisible.  While  the  Aristote- 
lian doctrine  held  sway  for  many  centuries  yet  the  notion  of  atoms 
was  revived  at  intervals.  Xate  in  the  seventeenth  century,  Boyle 
seems  to  have  looked  upon  chemical  combination  as  the  result  of 
atomic  association. 

Guided  by  these  early  speculations  as  to  the  constitution  of 


FUNDAMENTAL  PRINCIPLES  7 

matter  and  influenced  by  his  study  of  the  writings  of  Sir  Isaac 
Newton,  Dal  ton  seems  to  have  formed  a  mental  picture  of  the 
part  played  by  atoms  in  the  act  of  chemical  combination.  After 
a  few  carelessly  performed  experiments,  the  results  of  which  ac- 
corded with  his  preconceived  ideas,  he  formulated  his  atomic 
theory. 

According  to  this  theory  matter  is  composed  of  extremely  mi- 
nute, indivisible  particles  or  atoms.  Atoms  of  the  same  element 
are  all  of  equal  weight,  but  atoms  of  different  elements  have 
weights  proportional  to  their  combining  numbers.  Chemical 
compounds  are  formed  by  the  union  of  atoms  of  different  kinds. 
This  theory  offers  a  simple,  rational  explanation  of  the  laws  of 
chemical  combination. 

Since  a  chemical  compound  results  from  the  union  of  atoms, 
each  of  which  has  a  definite  weight,  its  composition  must  be  in- 
variable, —  which  is  the  law  of  definite  proportions.  Again, 
when  atoms  combine  in  more  than  one  proportion,  for  a  fixed 
weight  of  atoms  of  one  kind,  the  weights  of  the  other  species  of 
atoms  must  bear  to  each  other  a  simple  ratio,  since  the  atoms  are 
indivisible  units.  This  is  clearly  the  law  of  multiple  proportions. 

Finally,  the  law  of  combining  weights  is  seen  to  follow  as  a 
necessary  consequence  of  the  atomic  theory,  since  the  experimen- 
tally determined  combining  weights  bear  a  simple  relation  to  the 
relative  weights  of  the  atoms. 

At  the  time  when  Dalton  proposed  his  atomic  theory,  the 
number  of  facts  to  be  explained  was  comparatively  small,  but 
with  the  enormous  growth  of  the  science  of  chemistry  during  the 
past  century  and  with  the  vast  accumulation  of  data,  the  theory 
has  proved  capable  of  affording  adequate  representation  of  all  of 
the  facts,  and  has  opened  the  way  to  many  important  generaliza- 
tions. 

While  the  atomic  theory  has  played  a  very  important  part  in  the 
development  of  modern  chemistry,  and  while  we  recognize  that  it 
helps  to  clarify  our  thinking  and  enables  us  to  construct  a  mental 
image  of  tiny  spheres  uniting  to  form  a  chemical  compound,  yet  we 
must  not  forget  the  fact  that  these  atoms  are  purely  hypothetical. 

Faraday  has  said:  "Whether  matter  be  atomic  or  not,  this 


8  THEORETICAL  CHEMISTRY 

much  is  certain,  that  granting  it  to  be  atomic,  it  would  appear 
as  it  now  does."  Ostwald  believes  that  in  the  not  distant  future 
the  atomic  theory  will  be  abandoned  and  chemists  will  free  them- 
selves from  the  yoke  of  this  hypothesis,  relying  solely  upon  the 
results  of  experiment.  He  says:  "It  seems  as  if  the  adaptabil- 
ity of  the  atomic  hypothesis  is  near  exhaustion,  and  it  is  well 
to  realize  that,  according  to  the  lesson  repeatedly  taught  by  the 
history  of  science,  such  an  end  is  sooner  or  later  inevitable. " 

Combining  Weights  and  Atomic  Weights.  The  problem  of 
determining  the  relative  atomic  weights  of  the  elements  would  at 
first  sight  appear  to  be  a  very  simple  matter.  This  might  appar- 
ently be  accomplished  by  selecting  one  element,  say  hydrogen, 
it  being  the  lightest  known  element,  as  the  standard ;  a  compound 
of  hydrogen  and  another  element  may  then  be  analyzed  and  the 
amount  of  the  other  element  in  combination  with  one  part  by 
weight  of  hydrogen  determined.  This  weight  will  be  its  atomic 
weight  only  when  the  compound  contains  but  one  atom  of  each 
element.  To  determine  the  relative  atomic  weight,  therefore,  we 
must  know  in  addition  to  the  chemical  equivalent  of  the  element, 
the  number  of  atoms  present  in  the  compound.  For  example,  the 
analysis  of  water  shows  it  to  contain  8  parts  by  weight  of  oxygen 
to  1  part  by  weight  of  hydrogen;  the  chemical  equivalent  of 
oxygen  is,  therefore,  8,  and  if  water  contained  but  one  atom  of 
hydrogen  the  atomic  weight  of  oxygen  would  be  8.  It  can  be 
shown,  however,  that  water  contains  two  atoms  of  hydrogen 
and  one  atom  of  oxygen,  therefore,  the  atomic  weight  of  oxygen 
must  be  16.  It  is  evident,  therefore,  that  neither  the  analysis  nor 
the  synthesis  of  a  compound  is  sufficient  to  enable  us  to  determine 
the  number  of  atoms  of  an  element  combined  with  one  atom  of 
hydrogen.  We  shall  proceed  to  the  consideration  of  the  methods 
by  which  this  problem  may  be  solved. 

Gay-Lussac's  Law  of  Volumes.  Gay-Lussac  in  1808,  while 
studying  the  densities  of  gases  before  and  after  reaction,  announced 
the  following  law :  When  gases  combine  they  do  so  in  simple  ratios 
by  volume,  and  the  volume  of  the  gaseous  product  bears  a  simple 
ratio  to  the  volumes  of  the  reacting  gases  when  measured  under  like 
conditions  of  temperature  and  pressure.  Thus,  one  volume  of  hydro- 


FUNDAMENTAL  PRINCIPLES  9 

gen  combines  with  one  volume  of  chlorine  to  form  two  volumes 
of  hydrochloric  acid;  one  volume  of  oxygen  combines  with  two 
volumes  of  hydrogen  to  form  two  volumes  of  water  (vapor) ;  and 
one  volume  of  nitrogen  combines  with  three  volumes  of  hydrogen 
to  form  two  volumes  of  ammonia. 

In  a  previous  investigation,  Gay-Lussac  had  shown  that  all  gases 
behave  identically  when  subjected  to  changes  of  temperature  and 
pressure.  This  fact,  taken  together  with  the  simple  volumetric 
relation  just  enunciated  and  the  atomic  theory,  suggested  a  possible 
relation  between  the  number  of  ultimate  particles  in  equal  vol- 
umes of  different  gases. 

Berzelius  attempted  to  show  that  under  corresponding  condi- 
tions of  temperature  and  pressure,  equal  volumes  of  different 
gases  contain  the  same  number  of  atoms,  but  he  was  compelled 
to  abandon  the  assumption  as  untenable. 

Avogadro' s  Hypothesis.  It  remained  for  the  Italian  physicist, 
Avogadro,*  in  1811,  to  point  out  the  distinction  between  atoms 
and  molecules,  terms  which  had  been  used  almost  synonymously 
up  to  his  time.  He  denned  the  atom  as  the  smallest  particle 
which  can  enter  into  chemical  combination,  whereas  the  molecule 
is  the  smallest  portion  of  matter  which  can  exist  in  a  free  state. 
He  then  formulated  the  following  hypothesis  :f  Under  the  same 
conditions  of  temperature  and  pressure,  equal  volumes  of  all  gases 
contain  the  same  number  of  molecules.  This  hypothesis  has  been 
subjected  to  such  rigid  experimental  and  mathematical  tests  that 
its  validity  cannot  be  questioned. 

Avogadro's  Hypothesis  and  Molecular  Weights.  According 
to  Gay-Lussac  when  hydrogen  and  chlorine  combine  to  form  hydro- 
chloric acid,  one  volume  of  hydrogen  unites  with  one  volume  of 
chlorine  yielding  two  volumes  of  hydrochloric  acid. 

According  to  the  hypothesis  of  Avogadro,  the  number  of  mole- 
cules of  hydrochloric  acid  is  double  the  number  of  molecules  of 
hydrogen  or  of  chlorine,  and,  consequently,  each  molecule  of  the 
reacting  gases  must  contain  at  least  two  atoms.  If  we  take 
hydrogen  as  the  unit  of  our  system  of  atomic  weights,  its  molec- 

*  Jour,  de  Phys.,  73,  58  (1811). 

f  Ampere  advanced  nearly  the  same  hypothesis  in  1814. 


10  THEORETICAL  CHEMISTRY 

ular  weight  must  be  2.  It  is  convenient  to  express  molecular 
and  atomic  weights  in  terms  of  the  same  unit,  for  then  the  molec- 
ular weight  of  a  substance  will  be  simply  the  sum  of  the  weights 
of  the  atoms  contained  in  the  molecule.  The  determination  of 
the  approximate  molecular  weight  of  a  substance,  therefore,  re- 
solves itself  into  ascertaining  the  mass  of  its  vapor  in  grams  which, 
under  the  same  conditions  of  temperature  and  pressure,  will 
occupy  the  same  volume  as  2  grams  of  hydrogen. 

This  weight  is  called  the  gram-molecular  weight  or  the  molar 
weight  of  the  substance,  while  the  corresponding  volume  is  known 
as  the  gram-molecular  or  molar  volume.  It  is  nearly  the  same  for 
all  gases  and  at  0°  and  760  mm.  it  may  be  taken  equal  to  22.4 
liters.  The  molecular  weights  obtained  from  vapor  density  meas- 
urements are  approximate  only,  because  of  the  failure  of  most 
gases  and  vapors  to  obey  the  simple  gas  laws,  a  condition  essen- 
tial to  the  strict  applicability  of  Avogadro's  hypothesis. 

Atomic  Weights  from  Molecular  Weights.  While  vapor 
density  determinations  as  ordinarily  carried  out  do  not  give  exact 
molecular  weights,  it  is  an  easy  matter  to  arrive  at  the  true  values 
when  we  take  into  consideration  the  results  of  chemical  analysis. 
It  is  apparent  that  the  true  molecular  weight  must  be  the  sum  of 
the  weights  of  the  constituent  elements,  these  weights  being  exact 
multiples  or  submultiples  of  their  combining  proportions,  which 
proportions  have  been  determined  by  analysis  alone.  We  select, 
as  the  true  molecular  weight,  the  value  which  is  nearest  to  the 
approximate  molecular  weight  calculated  from  the  vapor  density 
of  the  substance.  For  example,  the  molecular  weight  of  ammonia, 
as  computed  from  its  vapor  density,  is  17.5  or,  in  other  words, 
17.5  grams  of  ammonia  occupy  the  same  volume  as  2  grams  of 
hydrogen,  measured  under  the  same  conditions  of  temperature 
and  pressure.  The  analysis  of  ammonia  shows  us  that  for  every 
gram  of  hydrogen,  there  are  present  4.67  grams  of  nitrogen. 
Hence  the  true  molecular  weight  must  contain  a  multiple  of  1  gram 
of  hydrogen  and  the  same  multiple  of  4.67  grams  of  nitrogen. 
The  problem  is,  to  find  what  integral  value  must  be  assigned  to 
x  in  the  expression,  x  (1  +  4.67),  in  order  that  it  may  give  the 
closest  approximation  to  17.5.  Clearly  if  x  =  3  the  value  of  the 


FUNDAMENTAL  PRINCIPLES 


11 


expression  becomes  17,  and  this  we  take  to  be  the  true  molecular 
weight.  This  gives  3  X  4.67  =  14  as  the  probable  atomic  weight 
of  nitrogen.  To  decide  whether  the  atomic  weight  of  nitrogen  is 
a  multiple  or  a  submultiple  of  14,  we  must  determine  the  molecu- 
lar weights  of  a  large  number  of  gaseous  or  vaporizable  compounds 
of  nitrogen  and  select  as  the  atomic  weight  the  smallest  quantity 
of  the  element  which  is  present  in  any  one  of  them. 

The  following  table  gives  a  list  of  seven  gaseous  compounds  of 
nitrogen  together  with  their  gram-molecular  weights,  and  the 
number  of  grams  of  the  element  in  the  gram-molecule. 


Compound. 

Gram-mol. 
Wt. 

Grams  Nitro- 
gen. 

Ammonia 

17 

14 

Nitric  oxide 

30 

14 

Nitrogen  peroxide 

46 

14 

Methyl  nitrate 

77 

14 

Cyanogen  chloride 

61.5 

14 

Nitrous  oxide 

44 

28 

Cyanogen 

52 

28 

It  will  be  observed  that  the  least  weight  of  nitrogen  entering 
into  a  gram-molecular  weight  of  any  of  these  compounds  is  14 
grams,  and,  therefore,  we  accept  this  value  as  the  atomic  weight 
of  the  element,  although  there  is  still  a  very  slight  chance  that 
in  some  other  compound  of  nitrogen  a  smaller  weight  of  the  ele- 
ment may  be  found.  We  shall  proceed  to  point  out  that  there 
are  methods  by  which  the  probable  values  of  the  atomic  weights 
may  be  checked. 

Specific  Heat  and  Atomic  Weight.  In  1819  the  French  chem- 
ists, Dulong  and  Petit,*  pointed  out  a  very  simple  relation  between 
the  specific  heats  of  the  elements  in  the  solid  state  and  their 
atomic  weights.  This  relation,  known  as  the  law  of  Dulong  and 
Petit,  is  as  follows :  The  product  of  the  specific  heat  and  the  atomic 
weight  of  the  solid  elements  is  constant.  The  value  of  this  constant, 
called  the  atomic  heat,  is  approximately  6.4.  A  little  reflection 
will  show  that  an  alternative  statement  of  this  law  is  that  the 

*  Ann.  Chim.  Phys.,  10,  395  (1819). 


12 


THEORETICAL  CHEMISTRY 


atoms  of  the  elements  in  the  solid  state  have  the  same  thermal  capac- 
ity. The  specific  heats,  atomic  weights  and  atomic  heats  of 
several  elements  are  given  in  the  subjoined  table. 


Element. 

At.  Wt. 

Sp.  Ht. 

At.  Ht. 

Lithium  

7 

0.940 

6.6 

Glucinum  

9 

0.410 

3.7 

Boron  (amorphous)  
Carbon  (diamond)    
Sodium  
Silicon  (crystalline)  

11 
12 
23 

28 

0.250 
0.140 
0.290 
0.160 

2.8 
1.7 
6.7 
4.5 

Potassium 

39 

0  166 

6.5 

Calcium 

40 

0.170 

6.8 

Iron 

56 

0  112 

6.3 

Copper.  . 

63 

0.093 

5.9 

Zinc  

65 

0.093 

6.1 

Silver  

108 

0.056 

6.0 

Tin.. 

119 

0.054 

6.5 

Gold 

197 

0  032 

6.3 

Mercury 

200 

0  032 

6.4 

It  is  truly  remarkable  that  elements  differing  as  greatly  as 
lithium  and  mercury  differ,  not  only  in  atomic  Weight  but  in 
other  properties  as  well,  should  have  identical  atomic  heats.  It 
will  be  observed  that  the  atomic  heats  of  boron,  silicon,  carbon 
and  glucinum  are  too  low.  This  departure  from  the  law  of  Dulong 
and  Petit  is  more  apparent  than  real,  for  in  the  statement  of  the 
law  there  is  no  specification  as  to  the  temperature  at  which  the 
specific  heat  should  be  determined.  The  specific  heats  of  all 
solids, vary  with  the  temperature,  this  variation  being  greater  in 
the  case  of  some  elements  than  in  that  of  others.  It  has  been 
shown  that  the  specific  heats  of  the  above  four  elements  increase 
rapidly  with  rise  of  temperature  and  approach  limiting  values. 
As  these  values  are  approached  the  product  of  specific  heat  and 
atomic  weight  approximates  more  and  more  closely  to  the  mean 
value  of  the  constant,  6.4. 

The  following  table  gives  the  values  obtained  by  Weber  *  for 
carbon  and  silicon. 


Pogg.  Ann.,  154,  367  (1875). 


FUNDAMENTAL  PRINCIPLES 
CARBON   (DIAMOND). 


13 


Temperature, 
degrees. 

Sp.  Ht. 

At.  Ht. 

-50 

0.0635 

0.76 

+10 

0.1128 

1.35 

85 

0.1765 

2.12 

206 

0.2733 

3.28 

607 

0.4408 

5.30 

806 

0.4489 

5.40 

985 

0.4589 

5.50 

CARBON   (GRAPHITE). 


Temperature, 
degrees. 

Sp.  Ht. 

At.  Ht. 

-50 

0.1138 

1.37 

+  10 

0.1604 

1.93 

61 

0.1990 

2.39 

202 

0.2966 

3.56 

642 

0.4454 

5.35 

822 

0.4539 

5.45 

978 

0.4670 

5.50 

SILICON. 


Temperature, 
degrees. 

Sp.  Ht. 

At.  Ht. 

-40 

0.136 

3.81 

+57 

0.183 

5.13 

129 

0.196 

5.50 

232 

0.203 

5.63 

It  is  evident  that  this  empirical  relation  can  be  used  to  deter- 
mine the  approximate  atomic  weight  of  an  element  when  its 
specific  heat  is  known,  thus 

6.4 

atomic  weight  = .  '  ,    —  • 

specific  heat 

The  law  of  Dulong  and  Petit  has  been  of  great  service  in  fixing 
and  checking  atomic  weights. 

About  twenty  years  after  the  law  of  Dulong  and  Petit  was 
formulated,  Neumann*  showed  that  a  similar  relation  holds  for 
*  Pogg.  Ann.,  23,  1  (1831). 


14 


THEORETICAL  CHEMISTRY 


compounds  of  the  same  general  chemical  character.  Neumann's 
law  may  be  stated  thus:  Similarly  constituted  compounds  in  the 
solid  state  have  the  same  molecular  heat.  Subsequently  Kopp  * 
pointed  out  that  the  thermal  capacity  of  the  atoms  is  not  appreciably 
altered  when  they  enter  into  chemical  combination,  or  in  other  words, 
the  molecular  heat  of  solid  compounds  is  an  additive  property, 
being  made  up  of  the  atomic  heats  of  the  constituent  elements. 

For  example,  the  specific  heat  of  PbBr2  is  0.054  and  its  molec- 
ular weight  is  366.8,  therefore,  the  molecular  heat  is  0.054  X  366.8 
=  19.9.  Since  there  are  three  atoms  in  the  molecule,  19.9  -f-  3 
=  6.6  is  their  average  atomic  heat,  a  value  in  excellent  agree- 
ment with  the  constant  in  the  law  of  Dulong  and  Petit.  Neu- 
mann's law  may  be  used  to  estimate  the  atomic  heats  of  elements 
which  cannot  be  readily  investigated  in  the  solid  state.  The 
following  table  gives  a  list  of  atomic  heats  of  elements  in  the  solid 
state  derived  by  means  of  Neumann's  law. 


Element. 

At.  Ht. 

Element. 

At.  Ht. 

Hydrogen  

2.3 

Carbon  

1.8 

Oxygen  

4.0 

Silicon  

4.0 

Fluorine  
Nitrogen 

5.0 
5  5 

Phosphorus  
Sulphur 

5.4 
5  4 

Isomorphism.  From  a  study  of  the  corresponding  salts  of 
phosphoric  and  arsenic  acids,  Mitscherlich  f  observed  that  they 
crystallize  with  the  same  number  of  molecules  of  water  and  are 
nearly  identical  in  crystalline  form,  it  being  possible  to  obtain 
mixed  crystals  from  solutions  containing  both  salts.  This  sug- 
gested to  Mitscherlich  a  line  of  investigation  which  resulted,  in 
1820,  in  the  establishment  of  the  law  of  isomorphism  which  bears 
his  name. 

This  law  may  be  stated  as  follows:  An  equal  number  of  atoms 
combined  in  the  same  manner  yield  the  same  crystal  form,  which  is 
independent  of  the  chemical  nature  of  the  atoms  and  dependent  upon 
their  number  and  position.  Thus,  when  one  element  replaces 

*  Lieb.  Ann.  (1864),  Suppl.,  3,  5. 

f  Ann.  Chim.  Phys.  (2),  14,  172  (1820). 


FUNDAMENTAL  PRINCIPLES 


15 


another  in  a  compound  without  changing  its  crystalline  form, 
Mitscherlich  assumed  that  one  element  has  displaced  the  other, 
atom  for  atom.  For  example,  having  two  isomorphous  substances, 
such  as  BaCl2.2  H20  and  BaBr2.2  H2O,  we  assume  that  the  brom- 
ine in  the  second  compound  has  replaced  the  chlorine  in  the  first 
and,  if  the  atomic  weights  of  all  of  the  elements  in  the  first  com- 
pound are  known,  then  it  is  evident  that  the  atomic  weight  of  the 
bromine  in  the  second  compound  can  be  easily  calculated.  This 
method  was  largely  used  by  Berzelius  in  fixing  atomic  weights  and 
in  checking  the  values  obtained  by  the  volumetric  method.  It 
should  be  remembered  that  the  converse  of  the  law  of  isomorphism 
does  not  hold,  since  elements  may  replace  each  other,  atom  for 
atom,  without  preserving  the  same  form  of  crystallization.  Many 
exceptions  to  the  law  have  been  pointed  out.  For  example, 
Mitscherlich  himself  showed  that  Na2SO4  and  BaMn208  are  iso- 
morphous and  yet  the  two  molecules  do  not  contain  the  same 
number  of  atoms.  Furthermore,  careful  measurements  of  the 
interfacial  angles  of  crystals  have  revealed  the  fact  that  sub- 
stances which  have  been  regarded  as  isomorphous  are  only  approx- 
imately so,  thus  the  interfacial  angles  of  the  apparently  isomorph- 
ous crystalline  salts  given  in  the  following  table  differ  appreciably. 


Salt. 

Interfacial  Angle. 

MgS04.7  H20 
ZnSO4.7  H2O 
NiSO4.7  H2O 

89°  26' 

88°  53' 
88°  56' 

Ostwald  has  suggested  that  the  term  homeomorphous  be  applied  to 
designate  substances  which  have  nearly  identical  form.  At  best 
the  principle  of  isomorphism  is  only  an  approximation  and  should 
be  employed  with  caution. 

Valence.  During  the  latter  half  of  the  nineteenth  century 
the  usefulness  of  the  atomic  theory  was  greatly  enhanced  by  the 
introduction  of  certain  assumptions  concerning  the  combining 
power  of  the  atoms.  These  assumptions,  constituting  the  so- 
called  doctrine  of  valence,  were  forced  upon  chemists  in  order 
that  a  satisfactory  explanation  might  be  offered  of  the  phenomenon 


16  THEORETICAL  CHEMISTRY 

of  isomerism.  A  consideration  of  the  following  formulas,  - 
HC1,  H20,  NH3,  CH4,  —  shows  that  the  power  to  combine  with 
hydrogen  increases  regularly  from  chlorine,  which  combines  with 
hydrogen,  atom  for  atom,  to  carbon,  one  atom  of  which  is  capable 
of  combining  with  four  atoms  of  hydrogen.  Either  hydrogen  or 
chlorine,  each  of  which  is  capable  of  combining  with  but  one 
atom  of  the  other,  may  be  taken  as  an  example  of  the  simplest 
kind  of  atom.  Any  element  like  hydrogen  or  chlorine  is  called 
a  univalent  element,  whereas  elements  similar  to  oxygen,  nitrogen 
and  carbon,  which  are  capable  of  combining  with  two,  three  or 
four  atoms  of  hydrogen,  are  called  bivalent,  trivalent  and  quadri- 
valent elements  respectively.  Most  elements  belong  to  one  or 
the  other  of  these  four  classes,  although  quinquivalent,  sexivalent 
and  septivalent  elements  are  known.  The  familiar  bonds  or  link- 
ages of  structural  formulas  are  graphic  representations  of  the 
valence  of  the  atoms  constituting  the  molecule.  This  useful  con- 
ception of  valence  has  made  possible  the  prediction  of  the  prop- 
erties of  many  compounds  before  they  have  been  discovered  in 
nature  or  in  the  laboratory. 

Atomic  Weights.  Among  the  first  to  recognize  the  importance 
of  Dalton's  atomic  theory  was  the  Swedish  chemist,  Berzelius. 
He  foresaw  the  importance  for  chemists  of  a  table  of  exact  atomic 
weights  and  in  1810  he  undertook  the  task  of  determining  the 
combining  weights  of  most  of  the  known  elements.  For  nearly 
six  years  he  was  engaged  in  determining  the  exact  composition  of 
a  large  number  of  compounds  and  calculating  the  combining 
weights  of  their  constituent  elements,  thus  compiling  the  first 
table  of  atomic  weights.  - 

Numerous  investigators  since  Berzelius  have  been  engaged  in 
this  important  work,  among  whom  should  be  mentioned  Stas, 
Marignac,  Morley  and  Richards.  On  two  occasions  special  stimu- 
lus was  given  to  such  investigations.  The  first  occasion  was  in 
1815  when  Prout  suggested  that  the  atomic  weights  of  the  elements 
are  exact  multiples  of  the  atomic  weight  of  hydrogen.  The  values 
obtained  by  Berzelius  were  incompatible  with  the  hypothesis  of 
Prout,  although  the  atomic  weights  of  several  of  the  elements 
differed  but  little  from  integral  values.  To  test  the  accuracy  of 


FUNDAMENTAL  PRINCIPLES  17 

this  hypothesis,  Stas  undertook  the  determination  of  the  atomic 
weights  of  several  elements  with  a  degree  of  accuracy  such  that 
his  maximum  experimental  error  was  less  than  the  difference 
between  the  atomic  weight  found  and  the  nearest  whole  number. 
This  important  series  of  investigations  disproved  Prout's  hypoth- 
esis as  originally  stated.  The  second  occasion  when  the  investi- 
gation of  atomic  weights  received  a  special  impulse  was  in  1869 
when  Mendeleeff  brought  forward  the  periodic  classification  of  the 
elements.  When  the  elements  were  arranged  in  the  order  of  their 
atomic  weights,  several  of  them  were  found  to  fall  in  groups  with 
which  their  chemical  and  physical  properties  did  not  correspond, 
and  Mendeleeff  asserted  that  in  these  cases  the  commonly  accepted 
atomic  weights,  were  erroneous.  This  led  to  the  careful  redeter- 
mination  of  the  atomic  weights  which  Mendeleeff  had  asserted 
to  be  faulty,  and  in  most  cases  his  predictions  were  confirmed. 

In  connection  with  the  precise  determination  of  atomic  weights, 
the  work  of  Cannizzaro  in  the  latter  part  of  the  nineteenth  century 
should  be  mentioned.  He  emphasized  the  importance  of  Avoga- 
dro's  law  as  the  basis  of  atomic  weight  determinations,  and  drew 
a  sharp  distinction  between  atomic  and  molecular  weights,  thus 
bringing  order  out  of  confusion  and  rendering  possible  the  present 
system  of  atomic  weights.  In  recent  times,  the  most  noteworthy 
investigations  in  this  field  are  those  of  Morley  on  the  combining 
ratio  of  hydrogen  and  oxygen,  and  the  determination  by  T.  W. 
Richards  and  his  co-workers  of  the  atomic  weights  of  a  large 
number  of  elements. 

International  Atomic  Weights.  Dalton  selected  hydrogen, 
the  lightest  known  element,  as  the  unit  of  his  system  of  combin- 
ing weights  of  the  elements,  but  Berzelius  pointed  out  that  this 
was  an  unwise  choice  since  but  relatively  few  of  the  elements  form 
stable  compounds  with  hydrogen.  He  proposed,  therefore,  that 
oxygen  should  be  taken  as  the  standard,  assigning  to  it  the  arbi- 
trary value  100.  This  proposal  of  Berzelius,  to  substitute  oxygen 
for  hydrogen  as  the  unit  of  atomic  weights,  did  not  receive  serious 
consideration  until  quite  recently  when  the  International  Com- 
mittee on  Atomic  Weights  took  the  matter  up  and,  after  careful 
deliberation,  decided  in  favor  of  the  oxygen  standard.  The 


18 


THEORETICAL  CHEMISTRY 


1917. 
INTERNATIONAL  ATOMIC  WEIGHTS. 


Aluminium  

A1 

O  =  16 

27.10 

Molybdenum  

Mo 

0  =  16 

96  00 

Antimony 

Sb 

120  20 

Neodymium 

Nd 

144  30 

Argon.  . 

A 

39.88 

Neon 

Ne 

20  20 

Arsenic  

As 

74.96 

Nickel 

Ni 

58  68 

Barium  

Ba 

137.37 

Niton  (radium  emanation). 

Nt 

222  40 

Bismuth  

Bi 

208.00 

Nitrogen  

N 

14  01 

Boron  

B 

11.00 

Osmium  

Os 

190  90 

Bromine  

Br 

79.92 

Oxygen  

o 

16  00 

Cadmium  

Cd 

112.40 

Palladium  

Pd 

106  70 

Caesium. 

Cs 

132  81 

Phosphorus 

p 

31  04 

Calcium  

Ca, 

40.07 

Platinum  

Pt 

195  20 

Carbon 

c 

12  005 

Potassium 

K 

39  10 

Cerium 

Ce 

140  25 

Praseodymium 

Pr 

140  90 

Chlorine 

Cl 

35.46 

Radium 

Ra 

226  00 

Chromium.  . 

Cr 

52  00 

Rhodium 

Rh 

102  90 

Cobalt  

Co 

58  97 

Rubidium 

Rb 

85  45 

Columbium  

Cb 

93.10 

Ruthenium 

Ru 

101  70 

Copper  

Cu 

63.57 

Samarium  .  .  . 

Sa 

150  40 

Dysprosium  

Dy 

162.50 

Scandium  . 

Sc 

44  10 

Erbium  

Fr 

167.70 

Selenium  

Se 

79  20 

Europium 

Eu 

152  00 

Silicon 

Si 

28  30 

Fluorine 

F 

19  00 

Silver 

Ae 

107  88 

Gadolinium  

Gd 

157.30 

Sodium  

Na 

23  00 

Gallium. 

Ga 

69  90 

Strontium 

Sr 

87  63 

Germanium 

Ge 

72  50 

Sulphur 

s 

32  06 

Glucinum.  . 

Gl 

9  10 

Tantalum 

Ta 

181  50 

Gold  

Au 

197  20 

Tellurium 

Te 

127  50 

Helium,  .  .    . 

He 

4  00 

Terbium 

Tb 

159  20 

Holmium  

Ho 

163  50 

Thallium.  . 

Tl 

204  00 

Hydrogen  

H 

1  008 

Thorium  

Th 

232  40 

Indium.  

In 

114  80 

Thulium  

Tm 

168  50 

Iodine  

I 

126  92 

Tin  

Sn 

118.70 

Iridium  
Iron  

Ir 
Fe 

193.10 

55  84 

Titanium  
Tungsten  

Ti 
W 

48.10 
184.00 

Krypton  
Lanthanum 

Kr 
La 

82.92 
139  00 

Uranium  
Vanadium 

U 

v 

238.20 
51  00 

Lead  

Pb 

207  20 

Xenon 

Xe 

130  20 

Lithium  
Lutecium  

Li 
Lu 

6.94 
175  00 

Ytterbium  (Neoytterbium) 
Yttrium  . 

Yb 
Yt 

173.50 

88.70 

Magnesium  

Mg 

24  32 

Zinc  

Zn 

65.37 

Manganese  
Mercury  

Mn 
Hg 

54.93 
200.60 

Zirconium  

Zr 

90.60 

FUNDAMENTAL  PRINCIPLES  19 

atomic  weight  of  oxygen  is  taken  as  16,  and  the  unit  to  which  all 
atomic  weights  are  referred  is  one-sixteenth  of  this  weight.  The 
atomic  weight  of  hydrogen  on  this  basis  is  1.008.  Aside  from  the 
fact  that  most  of  the  elements  form  compounds  with  oxygen  which 
are  suitable  for  analysis,  the  atomic  weights  of  more  of  the  ele- 
ments approximate  to  integral  values  when  oxygen  instead  of  hy- 
drogen is  used  as  the  standard. 

The  table  on  page  18  gives  the  values  of  the  atomic  weights 
as  published  by  the  International  Committee  on  Atomic 
Weights  for  1917. 


CHAPTER  II. 
CLASSIFICATION   OF  THE  ELEMENTS. 

Early  Attempts  at  Classification.  Many  attempts  were  made 
to  classify  the  elements  according  to  various  properties,  such  as 
their  acidic  or  basic  characteristics  or  their  valence.  In  all  of 
these  systems  the  same  elements  frequently  found  a  place  in  more 
than  one  group,  and  elements  bearing  little  resemblance  to  each 
other  were  classed  together.  The  early  attempts  at  classifica- 
tion based  upon  the  atomic  weights  of  the  elements  were  not 
successful  owing  to  the  uncertainty  as  to  the  exact  numerical 
values  of  these  constants. 

Prout's  Hypothesis.  In  1815,  W.  Prout,  an  English  physician, 
observed  that  the  atomic  weights  of  the  elements,  as  then  given, 
did  not  differ  greatly  from  whole  numbers  when  hydrogen  was 
taken  as  the  standard.  Hence  he  advanced  the  hypothesis  that 
the  different  elements  are  polymers  of  hydrogen.  As  has  already 
been  pointed  out  this  hypothesis  led  Stas  to  undertake  his  refined 
determinations  of  the  atomic  weights  of  silver,  lithium,  sodium, 
potassium,  sulphur,  lead,  nitrogen  and  the  halogens.  As  a  result 
of  his  investigations  he  says:  "I  have  arrived  at  the  absolute 
conviction,  the  complete  certainty,  so  far  as  is  possible  for  a  human 
being  to  attain  to  certainty  in  such  matter,  that  the  law  of  Prout 
is  nothing  but  an  illusion,  a  mere  speculation  definitely  contra- 
dicted by  experience."  Notwithstanding  the  fact  that  Prout 's 
hypothesis  as  originally  stated  was  thus  disproved  by  Stas,  it  still 
survived  in  a  modified  form  given  to  it  by  J.  B.  Dumas,  who  sug- 
gested that  one-half  of  the  atomic  weight  of  hydrogen  should  be 
taken  as  the  fundamental  unit.  When  Stas  showed  that  his 
experiments  excluded  this  possibility,  Dumas  suggested  that  the 
fundamental  unit  be  taken  as  one-quarter  of  the  atomic  weight 
of  hydrogen.  Having  begun  to  divide  and  subdivide,  there  was  no 
limit  to  the  process,  and  the  hypothesis  fell  into  disfavor,  although 

20 


CLASSIFICATION  OF  THE  ELEMENTS 


21 


the  belief  in  a  primal  element,  something  akin  to  the  protyle 
(-rrptary)  v\rj)  of  the  ancient  philosophers,  has  survived  and  in  modern 
times  has  reappeared  in  the  electron  theory. 

Dobereiner's  Triads.  About  1817  J.  W.  Dobereiner*  observed 
that  groups  of  three  elements  could  be  selected  from  the  list  of  the 
elements,  all  of  which  are  chemically  similar,  and  having  atomic 
weights  such  that  the  atomic  weight  of  the  middle  member  is  the 
arithmetical  mean  of  the  first  and  third  members  of  the  group. 
These  groups  of  three  elements  he  termed  triads.  In  the  following 
table  a  few  of  these  triads  are  given. 


Element. 

At.  Wt. 

Mean  atomic 
weight  of 
triads. 

Lithium.  . 

6  94 

) 

Sodium.  . 

23  00 

23  02 

Potassium  

39  10 

Calcium 

40  07 

) 

Strontium.    . 

87  63 

88  72 

Barium 

137  37 

Chlorine  

35.46 

Bromine  

79.92 

>     80  69 

Iodine 

126  92 

Sulphur  

32  07 

) 

Selenium 

79  2 

78  78 

Tellurium  *  
Phosphorus  

127.5 
31  04 

Arsenic  .    .    .    . 

74  96 

75  62 

Antimony  

120  2 

This  simple  relation,  first  pointed  out  by  Dobereiner,  is  clearly 
a  foreshadowing  of  the  periodic  law. 

The  Helix  of  de  Chancourtois.  The  idea  of  arranging  the 
elements  in  the  order  of  their  atomic  weights  with  a  view  to 
emphasizing  the  relationship  of  their  chemical  and  physical  prop- 
erties, seems  to  have  first  suggested  itself  to  M.  A.  E.  B.  de  Chan- 
courtois f  in  the  year  1862.  On  a  right-circular  cylinder  he  traced 


*  Pogg.  Ann.,  15,  301  (1825). 

t  Vis  Tellurique,  Classement  naturel  des  Corps  Simples. 


22 


THEORETICAL  CHEMISTRY 


what  he  termed  a  " telluric  helix"  at  a  constant  angle  of  45°  to  the 
axis.  On  this  curve  he  laid  off  lengths  corresponding  to  the 
atomic  weights  of  the  elements,  taking  as  a  unit  of  measure  a 
length  equal  to  one-sixteenth  of  a  complete  revolution  of  the 
cylinder.  He  then  called  attention  to  the  fact  that  elements  with 
analogous  properties  fall  on  vertical  lines  parallel  to  the  generatrix. 
Being  a  mathematician  and  a  geologist  he  did  not  express  himself 
in  such  terms  as  would  attract  the  attention  of  chemists  and  con- 
sequently his  work  remained  unnoticed  until  recent  times. 
The  Law  of  Octaves.  In  1864  J.  A.  R.  Newlands*  pointed 
out  that  if  the  elements  are  arranged  in  the  order  of  their  atomic 
weights,  the  eighth  element  has  properties  very  similar  to  the 
first;  the  ninth  to  the  second;  the  tenth  to  the  third;  and  so  on, 
or  to  employ  Newlands'  own  words :  "  The  eighth  element  starting  from 
a  given  one  is  a  kind  of  repetition  of  the  first,  like  the  eighth  note  of  an 
octave  in  music."  This  peculiar  relationship,  termed  by  New- 
lands  the  law  of  octaves,  is  brought  out  in  the  following  table. 


H 

Li 

Gl 

B 

C 

N 

0 

F 

Na 

Mg 

Al 

Si 

P 

S 

Cl 

K 

Ca 

Cr 

Ti 

Mn 

Fe 

Notwithstanding  the  fact  that  its  author  was  ridiculed  and  his 
paper  returned  to  him  as  unworthy  of  publication  in  the  proceed- 
ings of  the  Chemical  Society,  this  generalization  must  be  regarded 
as  the  immediate  forerunner  of  the  periodic  law. 

The  Periodic  Law.  Quite  independently  of  each  other  and 
apparently  in  ignorance  of  the  work  of  Newlands  and  de  Chan- 
courtois,  Mendeleeff  f  in  Russia  and  Lothar  Meyer  in  Germany, 
gamed  a  far  deeper  insight  into  the  relations  existing  between 
the  properties  of  the  elements  and  their  atomic  weights.  In  1869, 
Mendeleeff  wrote:  —  "When  I  arranged  the  elements  according 
to  the  magnitude  of  their  atomic  weights,  beginning  with  the 
smallest,  it  became  evident  that  there  exists  a  kind  of  periodicity 


*  Chem.  News,  10,  94  (1864),  Ibid.,  12,  83  (1865). 
t  Lieb.  Ann.  Suppl.,  8,  133  (1874). 


CLASSIFICATION  OF  THE  ELEMENTS  23 

in  their  properties.  I  designate  by  the  name  'periodic  law'  the 
mutual  relations  between  the  properties  of  the  elements  and  their 
atomic  weights;  these  relations  are  applicable  to  all  the  elements 
and  have  the  nature  of  a  periodic  function."  This  important 
generalization  may  be  briefly  stated  thus:  The  properties  of  the 
elements  are  periodic  functions  of  their  atomic  weights. 

The  original  table  of  Mendeleeff  has  been  amended  and  modified 
as  new  data  has  accumulated  and  new  elements  have  been  dis- 
covered. The  accompanying  table,  though  containing  several 
new  elements  and  an  entirely  new  group,  is  essentially  the  same  as 
that  of  Mendeleeff.  It  consists  of  nine  vertical  columns,  called 
groups,  and  twelve  horizontal  rows  termed  series  or  periods.  The 
second  and  third  periods  contain  eight  elements  each,  and  are 
known  as  short  periods,  while  in  the  fourth  series,  starting  with 
argon,  it  is  necessary  to  pass  over  eighteen  elements  before  another 
element,  krypton,  is  encountered  which  bears  a  close  resemblance 
to  argon:  such  a  series  of  nineteen  elements  is  called  a  long  period. 
The  entire  table  is  composed  of  two  short  and  five  long  periods, 
the  last  one  being  incomplete.  The  positions  of  the  elements  are 
largely  determined  by  their  chemical  similarity  to  those  in  the 
same  group,  the  hyphens  indicating  the  positions  of  undiscovered 
elements.  The  elements  in  Group  VIII,  presented  difficulties 
when  Mendeleeff  attempted  to  place  them  according  to  their 
atomic  weights  and  so  he  was  obliged  to  group  them  by  themselves. 
This  group  has  wittily  been  designated  as  "the  hospital  for  incur- 
ables." An  examination  of  the  table  shows  that  the  valence  of 
the  elements  toward  oxygen  progresses  regularly  from  Group  0, 
containing  elements  which  exhibit  no  combining  power,  up  to 
Group  VIII,  where  it  attains  a  maximum  value  of  eight  in  the 
case  of  osmium.  The  valence  toward  hydrogen  on  the  other  hand 
increases  regularly  from  Group  VII  to  Group  IV  in  which  the 
elements  are  quadrivalent. 


24 


THEORETICAL  CHEMISTRY 


V 
R04. 


S88888    22 

O  •— 


O 


t-  0 


<D 


s 


o 
N^       0 


= 


: 

s     j 


O 


60 


O 


(N 


00 


•sauag 


»d 

o 


? 


1     J      f       |      f 

OQ  S  H^  H^  ^ 


.          tc.2 


CLASSIFICATION  OF  THE  ELEMENTS  25 

The  formulas  of  the  typical  oxides  and  hydrides  of  the  elements 
in  the  several  groups  are  indicated  at  the  top  of  each  vertical 
column  in  the  table,  where  R  denotes  any  element  in  the  group. 
The  valence  of  elements  in  the  long  periods  are  apt  to  be  variable. 
The  elements  in  the  second  series  are  frequently  called  bridge 
elements,  since  they  bear  a  closer  relation  to  the  elements  in  the 
next  adjacent  group  than  they  do  to  any  other  members  of  the 
same  group  in  succeeding  series.  The  members  of  the  third  series 
are  styled  typical  elements,  because  they  exhibit  the  general  prop- 
erties and  characteristics  of  the  group.  Each  group  is  divided  into 
subgroups,  the  elements  on  the  right  and  left  sides  of  a  column 
forming  families,  the  members  of  which  are  more  closely  related 
than  are  all  of  the  elements  included  within  the  group.  In  other 
words  we  detect  a  kind  of  periodicity  within  each  group. 

In  any  given  series  the  element  with  lowest  atomic  weight 
possesses  the  strongest  basic  character.  Thus  we  find  the  strongly 
basic,  alkali  metals  on  the  left  side  of  the  table,  while  on  the  right 
side  are  the  acidic  elements  such  as  the  halogens  and  sulphur. 
In  fact,  the  strictly  non-metallic  elements  are  confined  to  the 
upper  right-hand  corner  of  the  table. 

Similarly,  as  we  pass  from  the  top  to  the  bottom  of  the  table, 
we  observe  a  progressive  change  in  the  base-forming  tendency  of 
the  elements;  i.e.,  as  the  atomic  weight  increases,  the  metallic 
character  of  the  elements  in  each  group  becomes  more  pronounced. 

Periodicity  of  Physical  Properties.  Lothar  Meyer,  as  has  been 
pointed  out,  discovered  the  periodic  relations  of  the  elements  at 
about  the  same  time  as  Mendeleeff.  His  table  differed  but  slightly 
from  that  already  given.  The  most  important  part  of  Meyer's  * 
work,  however,  was  in  pointing  out  that  various  physical  proper- 
ties of  the  elements  are  periodic  functions  of  their  atomic  weights. 
We  know  today  that  such  properties  as  specific  gravity,  atomic 
volume,  melting  point,  hardness,  ductility,  compressibility,  ther- 
mal conductivity,  coefficient  of  expansion,  specific  refraction,  and 
electrical  conductivity  are  all  periodic.  When  the  numerical 
values  of  these  properties  are  plotted  as  ordinates  against  their 
atomic  weights  as  abscissae,  we  obtain  wave-like  curves  similar  to 
*  Die  Modernen  Theorieri  der  Chemie. 


26 


THEORETICAL  CHEMISTRY 


those  shown  in  Fig.  2.  The  specific  heats  of  the  elements  are  an 
exception  to  the  general  rule.  According  to  the  law  of  Dulong  and 
Petit,  the  product  of  specific  heat  and  atomic  weight  is  a  constant, 
and  consequently  the  graphic  representation  of  this  relation  must 
be  an  equilateral  hyperbola. 

Applications  of  the  Periodic  Law.  Mendeleeff  pointed  out  the 
four  following  ways  in  which  the  periodic  law  could  be  employed: 
— (1)  The  classification  of  the  elements;  (2)  The  estimation  of  the 


Fig.  2. 

atomic  weights  of  elements;  (3)  The  prediction  of  the  properties 
of  undiscovered  elements;  and  (4)  The  correction  of  atomic 
weights. 

1.  Classification  of  Elements.    The  use  of  the  periodic  law  in 
this  direction  has  already  been  indicated.     It  is  without  doubt 
the  best  system  of  classification  known  and  is  to  be  ranked  among 
the  great  generalizations  of  the  science  of  chemistry. 

2.  Estimation   of  Atomic   Weights.     Because   of   experimental 
difficulties  it  is  not  always  possible  to  fix  the  atomic  weight  of  an 
element  by  determinations  of  the  vapor  densities  of  some  of  its 
compounds,  or  by  a  determination  of  its  specific  heat.     In  such 
cases  the  periodic  law  has  proved  of  great  value.     An  historic 


CLASSIFICATION  OF  THE  ELEMENTS  27 

example  is  that  of  indium,  the  equivalent  weight  of  which  was 
found  by  Winkler  to  be  37.8.  The  atomic  weight  of  the  element 
was  thought  to  be  twice  the  equivalent  weight  or  75.6.  If  this 
were  the  correct  value  it  would  find  a  place  in  the  periodic  table 
between  arsenic  and  selenium.  Clearly  there  is  no  vacancy  in 
the  table  at  this  point  and  furthermore  its  properties  are  not 
allied  to  those  of  arsenic  or  selenium.  Mendeleeff  proposed  to 
assign  to  it  an  atomic  weight  three  times  its  equivalent  weight  or 
113.4,  when  it  would  fall  between  cadmium  and  tin  in  the  table. 
This  would  bring  it  in  the  same  group  with  aluminium,  the  typical 
element  of  the  group,  to  which  it  bears  a  close  resemblance.  This 
suggestion  of  Mendeleeff's  was  confirmed  by  a  subsequent  deter- 
mination of  the  specific  heat  of  indium. 

3.  Prediction  of  Properties  of  Undiscovered  Elements.  At  the 
time  when  Mendeleeff  published  his  first  table  there  were  many 
more  vacant  spaces  than  exist  in  the  present  periodic  table.  He 
ventured  to  predict  the  properties  of  many  of  these  unknown 
elements  by  means  of  the  average  properties  of  the  two  neighbor- 
ing elements  in  the  same  series,  and  the  two  neighboring  elements 
in  the  same  subgroup.  These  four  elements  he  termed  atomic 
analogues.  The  undiscovered  elements  Mendeleeff  designated  by 
prefixing  the  Sanskrit  numerals,  eka  (one),  dwi  (two),  tri  (three), 
and  so  on,  to  the  names  of  the  next  lower  elements  of  the  sub- 
group. When  the  first  periodic  table  was  published  there  were 
two  vacancies  in  Group  III,  the  missing  elements  being  called  by 
Mendeleeff  eka-aluminium  and  eka-boron,  while  in  Group  IV 
there  was  a  vacancy  below  titanium,  the  missing  element  being 
called  eka-silicon.  The  subsequent  discovery  of  gallium,  scandium 
and  germanium,  with  properties  nearly  identical  with  those  pre- 
dicted for  the  above  hypothetical  elements,  served  to  strengthen 
the  faith  of  chemists  in  the  periodic  law.  The  following  table 
illustrates  the  accuracy  of  Mendeleeff's  prognostications:  in  it  is 
given  a  comparison  of  a  few  of  the  properties  of  the  hypothetical 
element,  eka-silicon,  as  predicted  by  Mendeleeff  in  1871,  and  the 
corresponding  observed  properties  of  germanium,  discovered  by 
Winkler  fifteen  years  later. 


28 


THEORETICAL  CHEMISTRY 


Eka-silicon,  Es. 


Germanium,  Ge. 


Atomic  weight,  72. 

Specific  gravity,  5.5. 

Atomic  volume,  13. 

Metal  dirty  gray,   and  on   ignition 

yields  a  white  oxide,  EsO2. 
Element    decomposes    steam    with 

difficulty. 
Acids  have  slight  action,  alkalies  no 

pronounced  action. 


Action  of  Na  on  EsO2  or  on  EsK2F6 
gives  metal. 

The  oxide  EsO2  refractory. 

Specific  gravity  of  EsO2,  4.7. 

Basic  properties  of  EsO2  less  marked 
than  TiO2  and  SnO2,  but  greater 
than  SiO2. 

Forms  hydroxide  soluble  in  acids, 
and  the  solutions  readily  decom- 
pose forming  a  metahydrate. 

EsCl4  a  liquid  with  a  b.p.  below  100° 
and  a  sp.  gr.  of  1.9  at  0°. 

EsF4  not  gaseous. 

Es  forms  a  compound  Es(C2H5)4  boil- 
ing at  160°,  and  with  a  sp.  gr.  0.96. 


Atomic  weight,  72.3. 
Specific  gravity,  5.47. 
Atomic  volume,  13.2. 
Metal  grayish-white,  and  on  igni- 
tion yields  a  white  oxide,  GeO2. 
Element  does  not  decompose  water. 

Metal  not  attacked  by  HC1,   but 
acted  upon  by  aqua  regia. 

Solutions  of  KOH  have  no  action. 
Oxidized  by  fused  KOH. 

Ge  obtained  by  reduction  of  GeO2 
with  C,  or  of  GeK2F6  with  Na. 

The  oxide  GeO2  refractory. 

Specific  gravity  of  GeO2,  4.703. 
Basic  properties  of  GeO2  feeble. 


Acids  do  not  ppt.  the  hydroxide 
from  dil.  alkaline  solutions,  but 
from  cone,  solutions,  acids  ppt. 
GeO  or  a  metahydrate. 

GeCl4  a  liquid  with  a  b.p.  of  86°, 
and  asp.  gr.  at  18°  of  1.887. 

GeF4.3  H2O  a  white  solid. 

Ge  forms  a  compound  Ge(C2H5)4 
boiling  at  160°  and  with  a  sp.  gr. 
slightly  less  than  water. 


4.  Correction  of  Atomic  Weights.  When  an  element  falls  in  a 
position  in  the  periodic  table  where  it  clearly  does  not  belong, 
suspicion  as  to  the  correctness  of  its  atomic  weight  is  immediately 
aroused.  Frequently  a  redetermination  of  the  atomic  weight  has 
revealed  an  error  which,  when  corrected,  has  resulted  in  assigning 
the  element  to  a  place  among  its  analogues.  Formerly  the 
accepted  atomic  weights  of  osmium,  iridium,  platinum  and  gold 
were  in  the  order 

Os  >  Ir  >  Pt  >  Au. 

But  from  analogies  existing  between  osmium,  ruthenium  and  iron 
and  the  disposition  of  the  preceding  members  of  Group  VIII, 
Mendeleeff  predicted  that  the  atomic  weights  were  in  error  and 
that  the  order  of  the  elements  should  be 
Os  <  Ir  <  Pt  <  Au. 


CLASSIFICATION  OF  THE  ELEMENTS  29 

Subsequent  atomic  weight  determinations  by  Seubert  substantiated 
MendeleefFs  prediction. 

Defects  in  the  Periodic  Law.  While  the  arrangement  of  the 
elements  in  the  periodic  table  is  on  the  whole  very  satisfactory, 
there  are  several  serious  defects  in  the  system  which  should  be 
pointed  out.  At  the  very  outset  there  is  difficulty  in  finding  a 
place  for  hydrogen  in  the  system.  The  element  is  univalent  and 
falls  either  in  Group  I,  with  the  alkali  metals,  or  in  Group  VII  with 
the  halogens.  While  the  element  is  electro-positive  it  cannot  be 
considered  to  possess  metallic  properties.  It  forms  hydrides  with 
some  of  the  metallic  elements  and  can  be  displaced  by  the  halo- 
gens from  organic  compounds.  These  facts  make  it  extremely 
difficult  to  decide  whether  hydrogen  should  be  placed  in  Group  I 
or  Group  VII.  The  idea  has  been  advanced  that  hydrogen  is 
the  only  known  member  of  the  first  series  of  the  periodic  table. 

These  hypothetical  elements  have  been  styled  proto-elements, 
the  successive  members  of  the  series  being,  proto-glucinum,  proto- 
boron  and  so  on  to  the  last  element  in  the  series,  proto-fluorine.  To 
find  a  suitable  location  for  the  rare-earth  elements  in  the  periodic 
system  is  another  difficulty  which  has  not  been  satisfactorily  met. 
Brauner  considers  that  these  elements  should  all  be  grouped 
together  with  cerium  (at.  wt.  =  140.25),  but  owing  to  our  limited 
knowledge  of  the  properties  of  these  elements  it  seems  better  to 
defer  attempting  to  place  them  for  the  present.  In  the  group  of 
non-valent  elements  the  atomic  weight  of  argon  is  distinctly  higher 
than  that  of  potassium  in  the  next  group.  There  can  be  little 
doubt  that  the  values  of  the  atomic  weights  are  correct  and  it 
is  evidently  impossible  to  interchange  the  positions  of  these  two 
elements  in  the  periodic  table,  since  argon  is  as  much  the  analogue 
of  the  rare  gases  as  potassium  is  of  the  alkali  metals.  A  similar 
discrepancy  occurs  with  the  elements,  tellurium  and  iodine.  The 
atomic  weight  of  the  former  element  is  appreciably  higher  than 
that  of  the  latter  and,  notwithstanding  the  attempts  of  numerous 
investigators  to  prove  tellurium  to  be  a  complex  of  two  or  more 
elements,  nothing  but  failure  has  attended  their  efforts.  Still 
another  anomaly  is  encountered  in  Group  VII,  where  manganese 
is  classed  with  the  halogen  family,  to  which  it  bears  much  less 


30  THEORETICAL  CHEMISTRY 

resemblance  than  it  does  to  chromium  and  iron,  its  two  immediate 
neighbors. 

As  has  already  been  mentioned,  Group  VIII  is  made  up  of  non- 
conformable  elements.  If  the  properties  of  the  elements  are 
dependent  upon  their  atomic  weights,  it  should  be  impossible  for 
several  elements  having  almost  identical  atomic  weights  and 
different  properties  to  exist,  and  yet  such  is  the  case  with  the 
elements  of  Group  VIII.  The  elements  copper,  silver  and  gold, 
while  not  closely  resembling  the  other  members  of  Group  VIII, 
are  much  more  closely  allied  to  them  than  to  the  alkali  metals 
with  which  they  are  also  classed.  Notwithstanding  its  imper- 
fections, the  periodic  law  must  be  regarded  as  a  truly  wonderful 
generalization  which  future  investigations  will  undoubtedly  show 
to  be  but  a  fragment  of  a  more  comprehensive  law. 


CHAPTER  III. 
THE  ELECTRON  THEORY. 

Conduction  of  Electricity  through  Gases.  Within  recent 
years  the  discovery  of  new  facts  relative  to  the  conduction  of 
electricity  through  gases  has  led  to  the  development  of  the  so- 
called  electron  or  corpuscular  theory  of  matter.  Under  ordinary 
conditions  gases  are  practically  non-conductors  of  electricity,  but 
when  a  sufficiently  great  difference  of  potential  is  established 
between  two  points  within  a  gas  it  is  no  longer  able  to  withstand 
tne  stress,  and  an  electric  discharge  takes  place  between  the  points. 
The  potential  necessary  to  produce  such  a  discharge  is  quite  hign, 


To  pump 
Fig.  3. 

several  thousand  volts  being  required  to  produce  a  spark  of  one 
centimeter  length  in  air  at  ordinary  pressures.  The  pressure  of 
the  gas  has  a  marked  effect  upon  the  character  of  the  discharge 
and  the  potential  required  to  produce  it.  If  we  make  use  of  a 
glass  vessel  similar  to  that  shown  in  Fig.  3,  the  effect  of  pressure 
on  the  nature  of  the  discharge  may  be  studied. 

This  apparatus  consists  of  a  straight  glass  tube  about  4  cm.  hi 
diameter  and  40  cm.  long,  into  the  ends  of  which  platinum  elec- 
trodes are  sealed.  To  the  side  of  the  vessel  a  small  tube  is  sealed 
so  that  connection  may  be  established  with  an  air-pump  and 
manometer.  If  the  electrodes  are  connected  with  the  terminals 
of  an  induction  coil  and  the  pressure  within  the  tube  be  gradually 
diminished,  the  following  changes  in  the  character  of  the  dis- 

31 


32 


THEORETICAL  CHEMISTRY 


charge  will  be  observed.  At  first  the  spark  becomes  more  uni- 
form and  then  broadens  out,  assuming  a  bluish  color.  When  a 
pressure  of  about  0.5  mm.  is 
reached,  the  negative  electrode 
or  cathode  will  appear  to  be 
surrounded  by  a  thin  luminous 
layer;  next  to  this  will  be  a 
dark  region,  known  as  the 


Crookes'  dark   space;  adjoining 
this  will  be  a  luminous  portion, 

called  the  negative  glow,  and  beyond  this  will  be  seen  another  dark 
region  which  is  frequently  referred  to  as  the  Faraday  dark  space. 
Between  the  Faraday  dark  space  and  the  positive  electrode  or 
anode  is  a  luminous  portion,  called  the  positive  column.  By  a  slight 
variation  of  the  current  and  pressure  the  positive  column  can  be 
caused  to  break  up  into  alternate  light  and  dark  spaces  or  strioe, 
the  appearance  of  which  is  dependent  upon  various  factors  such  as 


Fig.  5. 

the  nature  of  the  gas  and  the  size  of  the  tube.  If  we  use  a  modi- 
fication of  this  tube,  such  as  is  shown  in  Fig.  4,  and  diminish  the 
pressure  to  about  0.01  mm.,  a  new  phenomenon  will  be  observed. 
The  positive  column  will  vanish  and  the  walls  of  the  tube  opposite 
the  cathode  will  become  faintly  phosphorescent.  The  color  of 
the  phosphorescence  will  depend  upon  the  nature  of  the  glass: 
if  the  tube  is  made  of  soda  glass,  the  glow  will  be  greenish  yellow, 
while  with  lead  glass  the  phosphorescence  will  be  bluish.  The 
phosphorescence  is  due  to  the  bombardment  of  the  walls  of  the 


THE  ELECTRON  THEORY  33 

tube  by  very  minute  particles  projected  normally  from  the  cathode. 
These  streams  of  particles  are  called  the  cathode  rays. 

Some  Properties  of  Cathode  Rays.     The  following  are  among 
the  most  important  properties  of  the  cathode  rays :  — 

1.  The  cathode  rays  travel  in  straight  lines  normal  to  the  cathode: 
and  they  cast  shadows  of  opaque  objects  placed  in  their  path.     This 
property  may  be  demonstrated  by  means  of  the  apparatus  shown 
in  Fig.  5,  where  a  small  metallic  Maltese  cross  is  interposed  in  the 
path  of  the  rays,  a  distinct  shadow  being  cast  on  the  opposite  wall 
of  the  tube.     The  cross  may  be  hinged  at  the  bottom  so  that 
it  can  be  dropped  out  of  the  path  of  the  rays,  when  the  usual 
phosphorescence  will  be  obtained. 

2.  The  cathode  rays  can  produce  mechanical  motion.    By  means 


Fig.  o. 


of  the  apparatus  due  to  Sir  William  Crookes,  Fig.  6,  this  prop- 
erty of  the  cathode  rays  may  be  demonstrated.  Within  the 
vacuum  tube  is  placed  a  small  paddle  wheel  which  rolls  horizon- 
tally on  a  pair  of  glass  rails.  When  the  current  is  applied  to  the 
tube,  the  wheel  will  revolve,  moving  away  from  the  cathode.  By 
reversing  the  current,  the  wheel  will  stop  and  then  rotate  in  the 
opposite  direction  owing  to  the  reversal  of  the  direction  of  the 
cathode  stream. 

3.  The  cathode  rays  cause  a  rise  of  temperature  in  objects  upon 
which  they  fall.  In  the  tube  shown  in  Fig.  7,  the  anode  consists 
of  a  small  piece  of  platinum :  this  is  placed  at  the  center  of  curva- 
ture of  the  spherical  cathode.  After  pumping  down  to  the 
proper  pressure,  if  a  strong  discharge  be  sent  through  the  tube, 


34 


THEORETICAL  CHEMISTRY 


To  pump 


the  anode  will  begin  to  glow,  and  if  the  action  of  the  current  be 
continued  long  enough,  the  platinum  plate  may  be  rendered  in- 
candescent, thus  showing  the 
marked  heating  effect  of  the 
cathode  rays. 

4.  Many  substances  become 
phosphorescent  on  exposure  to 
the  cathode  rays.  If  the  cathode 
rays  be  directed  upon  different 
substances,  such  as  calc-spar, 

barium  platino-cyanide,  willemite,  scheelite  and  various  kinds 
of  glass,  beautiful  phosphorescent  effects  may  be  observed.  This 
phosphorescent  property  is  useful  in  observing  and  experimenting 
with  the  cathode  rays. 

5.  The  cathode  rays  can  be  deflected  from  their  rectilinear  path  by 
a  magnetic  field.     In   studying  the  magnetic   deviation   of  the 
cathode  rays  a  tube  similar  to  that  shown  in  Fig.    8  has  been 
found  very  satisfactory.     An  aluminium  diaphragm,  A,  pierced 


B 


Fig.  8. 

with  a  1  mm.  hole,  is  placed  in  front  of  the  cathode  while  at  the 
opposite  end  of  tube  is  placed  a  phosphorescent  screen,  D.  When 
the  discharge  takes  place  a  circular  phosphorescent  spot  will 
appear  on  D.  If  the  tube  be  placed  between  the  poles  of  an 
electromagnet,  the  phosphorescent  spot  will  move  at  right  angles 
to  the  direction  of  the  magnetic  field.  On  reversing  the  polarity 
of  the  magnet  the  spot  will  move  in  the  opposite  direction. 
Furthermore  the  direction  of  the  deflection  will  be  found  to  be 


THE  ELECTRON  THEORY 


35 


similar  to  that  produced  by  a  negative  charge  of  electricity  mov- 
ing in  the  same  direction  as  the  cathode  ray. 

6.  The  cathode  rays  can  be  deflected  from  their  rectilinear  path 
by  an  electrostatic  field.  The  same  tube  which  was  used  in  observ- 
ing the  magnetic  deflection  may  be  employed  in  studying  the 
effect  of  an  electrostatic  field.  .Two  insulated  metal  plates,  B 
and  C,  are  placed  on  opposite  sides  of  the  tube  and  parallel  to 
each  other.  When  the  tube  is  in  action,  if  a  difference  of  poten- 
tial of  several  hundred  volts  be  applied  to  the  plates,  the  phos- 
phorescent spot  on  D  will  be  found  to  move,  the  direction  of  the 


Elect. 


To  Pump 


Fig.  9. 


motion  being  the  same  as  that  of  a  negatively  charged  body  under 
the  influence  of  an  electrostatic  field.  Reversal  of  the  field  causes 
the  phosphorescent  spot  to  move  to  the  opposite  side  of  the 
screen. 

7.  The  cathode  rays  carry  a  negative  charge.  Probably  the  most 
important  characteristic  of  the  cathode  rays  is  their  ability  to 
carry  a  negative  charge.  While  the  magnetic  and  electrostatic 
deviation  of  the  rays  made  this  fact  more  than  probable,  it  re- 
mained for  Perrin  to  demonstrate  that  a  negative  electrification 
accompanies  the  cathode  stream.  A  modification  of  Perrin's 
apparatus  due  to  J.  J.  Thomson  is  shown  in  Fig.  9.  It  con- 


36  THEORETICAL  CHEMISTRY 

sists  of  a  spherical  bulb  to  which  is  sealed  a  smaller  bulb  and  a  long 
side  tube.     The  small  bulb  contains  the  cathode  C  and  the  anode 

A.  The  anode  consists  of  a  tight-fitting  brass  plug  pierced  by  a 
central  hole  of  small  diameter.     The  side  tube,  which  is  out  of 
the  direct  range  of  the  cathode  rays,  contains  two  coaxial  metallic 
cylinders  insulated  from  each  other,  each  being  perforated  with  a 
narrow  transverse  slit:   D  is  earth-connected  and  B  is  connected 
with  an  electrometer  by  means  of  the  rod  F.     When  the  tube  has 
been  pumped  down  to  the  proper  pressure  for  the  production  of 
cathode  rays,  a  phosphorescent  spot  will  appear  at  E  directly 
opposite  the  cathode  C.     Upon  testing  B  for  possible  electrifica- 
tion by  means  of  the  electrometer,  it  will  be  found  to  be  uncharged. 
If  the  cathode  stream  be  deflected  by  means  of  a  magnet  so  that 
the  rays  fall  upon  B,  a  sudden  charging  of  the  electrometer  will  be 
observed,  proving  that  B  is  becoming  electrified.     Upon  deflect- 
ing the  rays  still  further  so  that  they  are  no  longer  incident  upon 

B,  the  accumulation  of  charge  will  immediately  cease.     If  the 
electrometer  be  tested  for  polarity,  it  will  be  found  to  be  negatively 
charged,  thus  proving  the  charge  carried  by  the  cathode  rays  to 
be  negative. 

8.  The  cathode  rays  can  penetrate  thin  sheets  of  metal.     In  1894 
Lenard  constructed  a  vacuum  tube  fitted  with  an  aluminium 
window  opposite  the  cathode.     He  showed  that  the  cathode  rays 
passed  through  the  aluminium  and  are  absorbed  by  different  sub- 
stances outside  of  the  tube,  the  absorption  varying  directly  with 
the  density  of  the  substance. 

9.  The  cathode  rays  when  directed  into  moist  air  cause  the  forma- 
tion of  fog.     This  phenomenon  has  been  shown  by  C.  T.  R.  Wilson 
to  be  due  to  the  minute  particles  in  the  cathode  stream  acting  as 
nuclei  upon  which  the  water  vapor  can  condense. 

Velocity  of  the  Cathode  Particle.  Since  the  cathode  rays 
consist  of  minute,  negatively-charged  particles  which  can  be 
deflected  by  a  magnetic  and  an  electrostatic  field,  it  is  possible 
to  measure  their  speed  and  to  compute  the  ratio  of  the  mass  of  a 
particle  to  its  charge.  The  special  form  of  tube  shown  in 
Fig.  10  was  devised  for  the  purpose  by  J.  J.  Thomson.  It  consists 
of  a  glass  tube  about  60  cm.  in  length,  furnished  with  a  flat  cir- 


THE  ELECTRON  THEORY 


37 


cular  cathode,  C,  and  an  anode,  A,  in  the  form  of  a  cylindrical 
brass  plug  about  2.5  cm.  in  length,  pierced  by  a  central  hole  1  mm. 
in  diameter.  Another  brass  plug,  B,  is  placed  about  5  cm.  away 
from  A,  the  two  holes  being  in  exactly  the  same  straight  line,  so 
that  a  very  narrow  bundle  of  rays  may  pass  along  the  axis  of  the 
tube  and  fall  upon  the  phosphorescent  screen  at  the  opposite 
end  of  the  tube.  Upon  this  screen  is  a  millimeter  scale,  SS'. 
Two  parallel  plates,  D  and  E,  are  sealed  into  the  tube  for  the  pur- 
pose of  establishing  an  electrostatic  field.  When  the  tube  is 
connected  with  an  induction  coil  or  other  source  of  high-potential, 
a  phosphorescent  spot  will  appear  at  F.  If  a  strong  magnetic 
field  be  applied,  the  lines  of  force  being  at  right  angles  to  the  plane 


Fig.  10. 

of  the  diagram,  the  rays  will  be  deflected  vertically  and  the  spot 
on  the  screen  will  move  from  F  to  G. 

Let  H  denote  the  strength  of  the  magnetic  field  and  let  m,  e 
and  v  represent  respectively  the  mass,  charge  and  velocity  of  a 
cathode  particle.  A  magnetic  field,  H,  acting  at  right  angles  to 
the  line  of  flight  of  the  cathode  particle  will  exert  a  force, 
Hev,  which  will  tend  to  deflect  the  particle  from  a  rectilinear 
path.  This  force  must  be  equal  to  the  centrifugal  force  of  the 
moving  particle  acting  outwards  along  its  radius  of  curvature. 
Therefore 

mv* 


Hev  =  -  — , 


or 


(D 


38  THEORETICAL  CHEMISTRY 

Since  H  and  r  can  both  be  measured,  the  ratio,  — ,  can  be  deter- 

6 

mined.  Now  if  a  difference  of  potential  be  established  between 
D  and  E,  and  the  lines  of  force  in  the  electrostatic  field  have  the 
proper  direction,  it  will  be  possible  to  alter  the  strength  of  the  field 
so  as  to  just  counterbalance  the  effect  of  the  magnetic  field,  and 
bring  the  phosphorescent  spot  back  to  F  again.  Under  these 
conditions,  if  X  denotes  the  strength  of  the  electrostatic  field,  we 
have 

Xe  =  Hev, 
or 

-i-  (2) 

Since  X  and  H  can  both  be  measured,  v  can  be  calculated,  and  by 
introducing  the  value  so  obtained  into  equation  (1),  the  ratio  e/m 
can  be  evaluated.  By  this  method  the  average  value  of  v  has 
been  found  to  be  2.8  X  109  cm.  per  second,  while  1.7  X  107  is 
the  mean  value  of  a  large  number  of  determinations  of  the  ratio 
e/m. 

Comparison  of  the  Ratio  of  Charge  to  Mass  for  the  Cathode 
Particle  with  that  for  the  Ion  in  Electrolysis.  The  ratio  of  the 
charge  carried  by  an  ion  in  electrolysis  to  its  mass  can  be  easily 
computed.  Thus  it  may  be  shown  that  the  ratio  of  the  charge  E, 
of  the  hydrogen  ion  to  its  mass,  M ,  in  electrolysis  is  about  1  X  104 
C.G.S.  units  or 

ET 

-JTJ.  =  1  X  104  approximately. 

The  mass  of  the  hydrogen  ion  may  be  considered  to  be  identical 
with  that  of  the  hydrogen  atom,  the  lightest  atom  known.  Com- 
paring the  value  of  e/m  for  the  cathode  particle  with  the  value  of 
E/M  for  the  hydrogen  ion  in  electrolysis,  it  is  evident  that  the 
former  is  about  1700  times  greater  than  the  latter. 

Charge  Carried  by  the  Cathode  Particle.  Until  the  value 
of  the  charge  carried  by  the  cathode  particle  has  been  determined, 
it  is  clearly  impossible  to  compute  its  mass.  Thus,  if  we  consider 


THE  ELECTRON  THEORY  39 

the  last  statement  of  the  preceding  paragraph,  which  may  be 
formulated  as  follows :  — 

e/m  :E/M  ::  1700  :  1, 

the  proportion  will  remain  unaltered  whether  m  =  M/1700  and 
e  =  E,  or  e  =  1700  and  m  =  M .  The  method  employed  to  deter- 
mine the  charge  carried  by  a  cathode  particle  is  too  complicated 
for  a  detailed  description  in  this  place;  merely  the  general  outline 
will  be  given.  Upon  suddenly  expanding  a  volume  of  saturated 
water  vapor,  its  temperature  is  lowered,  and  a  cloud  forms,  each 
particle  of  dust  present  serving  as  a  nucleus  for  a  fog  particle.  If 
sufficient  time  be  allowed  for  the  mist  to  settle  and  the  vapor  to 
become  saturated  again,  a  repetition  of  the  preceding  process  will 
result  in  the  formation  of  less  mist,  owing  to  the  presence  of  fewer 
dust  particles.  By  repeating  the  operation  enough  tunes  the 
space  may  be  rendered  dust  free.  As  has  already  been  pointed 
out,  cathode  particles  serve  as  nuclei  for  the  condensation  of 
water  vapor,  their  function  being  similar  to  that  of  dust  particles. 
It  has  been  shown  by  Sir  George  G.  Stokes  that  if  a  drop  of  water 
of  radius  r,  be  allowed  to  fall  through  a  gas  of  viscosity  77,  then  the 
velocity  with  which  the  drop  falls  is  given  by  the  equation 

.-;.£ 

where  g  is  the  acceleration  due  to  gravity.  The  viscosity  of  air 
at  any  temperature  being  known,  a  cloud  can  be  produced  in  an 
appropriate  chamber  by  expansion  of  water  vapor  in  the  presence 
of  cathode  particles  and  the  speed,  v,  with  which  the  cloud  falls 
can  be  measured,  and  hence  r  can  be  calculated  by  means  of  equa- 
tion (3).  If  m  is  the  total  mass  of  the  cloud  and  n  is  the  number  of 
drops  per  cubic  centimeter,  then 

m  =  4/3  rnrr*  (density  of  water  =  1). 

From  a  simple  application  of  thermodynamics  m  may  be  deter- 
mined. Knowing  the  values  of  m  and  r,  the  number  of  drops 
in  the  cloud,  n,  which  is  the  same  as  the  number  of  cathode  par- 
ticles, can  be  calculated.  It  is  a  simple  matter  to  measure  the 
total  charge  in  the  expansion  chamber,  and  dividing  this  by  the 
total  number  of  charged  particles,  gives  the  charge  carried  by  a 


40  THEORETICAL  CHEMISTRY 

single  particle.  The  latest  determinations  of  J.  J.  Thomson  show 
this  to  be  3.4  X  ~10  electrostatic  unit.  This  is  practically 
identical  with  the  calculated  value  of  the  charge  on  the  hydro- 
gen ion  in  electrolysis,  or  e  =  E  and  therefore  m  =  Af/1700;  the 
mass  of  the  cathode  particle  is  1/1700  of  the  mass  of  the  atom 
of  hydrogen.  The  cathode  particle  has  the  smallest  mass  yet 
known  and  has  been  called  the  corpuscle  or  electron. 

An  ingenious  modification  of  the  foregoing  method  devised  by 
Millikan,*  has  made  it  possible  to  determine  e  with  extreme 
accuracy.  Millikan  gives  as  the  mean  of  a  large  number  of  deter- 
minations of  e,  the  following  value  which  he  states  is  in  error  by 
less  than  0.1  per  cent: 

e  =  4.4775  X  10~10  electrostatic  units. 

For  purposes  of  comparison,  the  following  values  of  e  obtained  by 
other  investigators  using  different  experimental  methods  are  here 
given :  — 

(a)  Making  use  of  available  data  on  radiant  energy,  Planck 
calculated  e  =  4.69  X  1Q-10. 

(6)  By  counting  the  number  of  scintillations  produced  by  a 
known  weight  of  polonium  and  measuring  the  total  charge,  Regener 
found  e  =  4.79  X  1Q-10. 

(c)  By  counting  the  number  of  a-particles  escaping  from  a  given 
amount  of  radium  bromide  and  measuring  the  total  charge, 
Rutherford  and  Geiger  calculated  e  =  4.65  X  10~10. 

The  Avogadro  Constant.  The  actual  number  of  molecules  in 
one  gram-molecule  or  the  actual  number  of  atoms  in  one  gram- 
atom  of  a  gas  is  called  the  Avogadro  Constant.  The  most  accurate 
method  for  the  calculation  oi  this  constant  involves  the  elementary 
charge  of  electricity,  e. 

The  quantity  of  electricity  carried  by  one  gram  equivalent  in 
electrolysis  has  been  found  to  be  96,500  coulombs  (see  p.  390). 
This  quantity,  known  as  the  faraday,  and  commonly  designated 
by  F,  bears  the  same  relation  to  e  that  the  actual  weight  of  an  atom 
bears  to  the  number  expressing  its  atomic  weight.  The  relation 

*  Phil.  Mag.,  6,  19,  209  (1910);  Phys.  Rev.,  39,  349  (1911);  Trans.  Am. 
Electrochem.  Soc.,  21,  185  (1912). 


THE  ELECTRON  THEORY  41 

between  the  Avogadro  constant  N  and  the  ionic  and  electronic 
charges  F  and  e,  is  given  by  the  equation 

F 

N  =  -- 
e 

Substituting  the  above  values  in  this  equation  and  converting 
coulombs  into  electrostatic  units,  we  have 

96500X3X10* 
'   4.4775  X  10-10 

Other  Sources  of  Electrons.  Electrons  may  be  produced  by 
other  agents  than  cathode  rays.  Thus,  electrons  are  emitted  by 
radium  and  other  radioactive  substances,  by  metals  and  amalgams 
under  the  influence  of  ultra-violet  light,  and  also  by  gas  flames 
charged  with  the  vapors  of  salts.  It  has  been  shown  that  from 
whatever  source  an  electron  is  derived,  the  value  of  the  ratio  e/m 
remains  constant.  This  interesting  fact  has  led  Thomson  to  sug- 
gest that  the  electron  may  be  regarded  as  "one  of  the  bricks  of 
which  the  atom  is  built  up." 

Before  entering  upon  a  discussion  of  modern  views  of  atomic 
structure,  however,  it  will  be  necessary  to  summarize  very  briefly 
some  of  the  salient  facts  of  radiochemistry. 


CHAPTER   IV. 
RADIOACTIVITY. 

Discovery  of  Radioactivity.  The  first  radioactive  substance 
was  discovered  by  Henri  Becquerel  *  in  1896.  It  had  been  shown 
by  Roentgen  in  the  previous  year  that  the  bombardment  of  the 
walls  of  a  vacuum  tube  by  the  cathode  stream,  gives  rise  to  a  new 
type  of  rays,  which,  because  of  their  puzzling  characteristics,  he 
called  X-rays.  The  portion  of  the  tube  where  these  rays  originate 
was  observed  to  fluoresce  brilliantly,  and  it  was  at  once  assumed 
that  this  fluorescence  might  be  the  cause  of  the  new  type  of  radia- 
tion. 

Many  substances  were  known  to  fluoresce  under  the  stimulus  of 
the  sun's  rays,  and  it  was  natural,  in  the  light  of  Roentgen's  dis- 
covery, that  all  substances  which  exhibit  fluorescence  should  be 
subjected  to  careful  examination.  Among  those  who  became  inter- 
ested in  these  phenomena  was  Becquerel.  He  studied  the  action" of 
a  number  of  fluorescent  substances,  among  which  was  the  double 
sulphate  of  potassium  and  uranium.  This  salt,  after  exposure  to 
sunlight,  was  found  to  emit  a  radiation  capable  of  affecting  a  care- 
fully protected  photographic  plate.  Further  investigation  proved 
that  the  fluorescence  had  nothing  to  do  with  the  photographic 
action,  since  both  uranous  and  uranic  salts  were  found  to  exert 
similar  photographic  action,  notwithstanding  the  fact  that  uranous 
salts  are  not  fluorescent.  The  photographic  activity  of  both 
uranous  and  uranic  salts  was  found  to  be  proportional  to  their 
content  of  uranium.  Becquerel  also  showed  that  preliminary 
stimulation  by  sunlight  was  wholly  unnecessary.  Uranium  salts 
which  had  been  kept  in  the  dark  for  years  were  found  to  be  just  as 
active  as  those  which  had  been  recently  exposed  to  brilliant  sun- 
light. The  properties  of  the  rays  emitted  by  uranium  salts  differ 
in  many  respects  from  those  of  the  X-rays.  The  rate  of  emission 
of  the  uranium  rays  remains  unaltered  at  the  highest  or  the  lowest 

*  Compt.  rendus,  122,  420  (1896). 
42 


RADIOACTIVITY  43 

obtainable  temperatures.  The  entire  behavior  of  these  salts  justi- 
fies the  conclusion  that  the  continuous  emission  of  penetrating 
rays  is  a  specific  property  of  the  element  uranium  itself.  This 
property  of  spontaneously  emitting  radiations  capable  of  penetra- 
ting substances  opaque  to  ordinary  light  is  called  radioactivity. 

Discovery  of  Radium.  Shortly  after  the  discovery  of  the 
radioactivity  of  uranium,  the  element  thorium  and  its  compounds 
were  also  found  to  be  radioactive.  As  a  result  of  a  systematic 
examination  by  Mme.  Curie  *  of  minerals  known  to  contain 
uranium  or  thorium,  it  was  learned  that  many  of  these  were  much 
more  radioactive  than  either  uranium  or  thorium  alone.  Thus, 
pitchblende,  one  of  the  principal  ores  of  uranium,  was  found  to 
be  four  times  more  active  than  uranium  alone,  and  chalcolite,  a 
phosphate  of  copper  and  uranium,  was  found  to  be  at  least  twice 
as  active  as  uranium.  On  the  other  hand,  when  a  specimen  of 
artificial  chalcolite,  prepared  in  the  laboratory  from  pure  materials, 
was  examined,  its  activity  was  found  to  be  proportional  to  the 
content  of  uranium.  Mme.  Curie  concluded  from  this  result  that 
natural  chalcolite  and  pitchblende  must  contain  a  minute  amount 
of  some  substance  much  more  active  than  uranium. 

With  the  assistance  of  her  husband,  Mme.  Curie  undertook  the 
task  of  separating  this  unknown  substance  from  pitchblende. 
Pitchblende  is  an  extremely  complex  mineral  and  its  systematic 
chemical  analysis  calls  for  skill  and  patience  of  a  high  order.  With- 
out "entering  into  details  as  to  the  analytical  procedure,  it  must 
suffice  here  to  state  the  results  obtained.  Associated  with  bismuth, 
a  very  active  substance  was  discovered,  to  which  Mme.  Curie  gave 
the  name  polonium  in  honor  of  her  native  land,  Poland.  In  like 
manner,  an  extremely  active  substance  was  found  associated  with 
barium  in  the  alkaline  earth  group.  The  substance  was  called 
radium  because  of  its  great  radioactivity. 

While  the  isolation  of  pure  polonium  is  extremely  difficult  and, 
while  sufficient  quantities  have  not  been  obtained  to  permit  de- 
terminations of  its  physical  properties,  the  isolation  of  radium 
in  relatively  large  amounts  is  readily  accomplished.  The  pure 
bromides  of  radium  and  barium  are  prepared  together  and  the 
*  Compt.  rendus,  126,  1101  (1898). 


44  THEORETICAL  CHEMISTRY 

two  salts  are  then  separated  by  a  series  of  fractional  crystalliza- 
tions. That  the  salts  of  barium  and  radium  are  very  similar  in 
chemical  properties  is  shown  by  the  fact  that  they  separate  to- 
gether from  the  same  solution.  The  atomic  weight  of  radium  has 
been  determined  by  several  investigators,  the  accepted  value  being 
226.  It  is  thus,  with  the  exception  of  uranium,  the  heaviest  known 
element. 

In  1910,  Mme.  Curie  *  succeeded  in  obtaining  metallic  radium. 
It  is  a  metal  possessing  a  silvery  luster,  dissolving  in  water  with 
energetic  evolution  of  hydrogen  and  tarnishing  rapidly  in  air  with 
the  formation  of  the  nitride. 

It  is  estimated  that  one  ton  of  pitchblende  contains  approxi- 
mately 0.2  gram  of  radium. 

Other  Radioactive  Substances.  Shortly  after  the  discovery 
of  radium  and  polonium  by  the  Curies,  Debierne  f  discovered 
another  radioactive  element  in  pitchblende.  This  element,  which 
he  named  actinium,  was  found  associated  with  the  iron  group  in 
the  course  of  the  analysis  of  the  mineral. 

In  1906,  Boltwood  J  discovered  in  pitchblende,  and  in  several 
other  uranium  minerals,  the  presence  of  still  another  radioelement 
which  he  named  ionium.  Ionium  is  much  more  active  than 
thorium  to  which  it  bears  such  a  close  resemblance  that  the  two 
elements  cannot  be  separated  from  each  other. 

The  lead  which  is  obtained  from  uranium  and  thorium  minerals 
is  found  to  be  slightly  radioactive,  the  activity  being  attributed 
to  the  presence  of  a  small  proportion  of  a  constituent  called  radio- 
lead,  the  chemical  properties  of  which  resemble  those  of  ordinary 
lead.  It  is  interesting  to  note  that  recent  determinations  by 
Richards  §  of  the  atomic  weight  of  lead  obtained  from  different 
sources,  reveal  differences  greater  than  the  possible  experimental 
errors  of  the  determinations.  Thus,  the  values  of  the  atomic 
weight  of  lead  from  pitchblende  and  from  thorite  were  found  to 
be  206.40  and  208.4  respectively,  while  the  value  of  the  atomic 
weight  of  ordinary  lead  is  207.15. 

*  Compt.  rendus,  151,  523  (1910). 

t  Compt.  rendus,  129,  593  (1899). 

J  Am.  Jour.  Sci.,  22,  537  (1906). 

§  Jour.  Am.  Chem.  Soc.,  36,  1329  (1914). 


RADIOACTIVITY  45 

About  thirty  other  radioactive  substances  have  been  separated 
and  many  of  their  properties  have  been  determined.  All  of  these 
radio-elements  have  been  shown  to  be  the  lineal  descendants  of  one 
or  the  other  of  the  two  parent  elements,  uranium  or  thorium. 

lonization  of  Gases.  The  radiations  emitted  by  radioactive 
substances  have  the  power  of  rendering  the  air  through  which  they 
pass  conductors  of  electricity.  To  account  for  this  action,  Thom- 
son and  Rutherford  formulated  the  theory  of  gaseous  ionization. 
According  to  this  theory,  which  has  since  been  experimentally 
confirmed,  the  radiations  break  up  the  components  of  the  gas  into 
positive  and  negative  carriers  of  electricity  called  ions.  If  two 
parallel  metal  plates  are  connected  to  the  terminals  of  a  battery 
and  a  radioactive  substance  is  placed  between  them,  the  air  will  be 
ionized  and,  owing  to  the  movement  of  the  positive  and  negative 
ions  toward  the  plates  of  opposite  sign,  an  electric  current  will 
pass  between  the  plates.  If  the  electric  field  is  weak,  the  mutual 
attraction  between  the  positive  and  negative  ions  will  cause  many 
of  them  to  recombine  before  reaching  the  plates  and  the  resulting 
current  will  be  small.  As  the  strength  of  the  field  is  increased,  the 
greater  will  be  the  speed  of  the  ions  toward  the  plates  and  the 
smaller  will  become  the  tendency  to  recombination.  Ultimately, 
with  increasing  strength  of  field  all  of  the  ions  will  be  swept  to  the 
plates  as  fast  as  they  are  formed  and  the  ionization  current  will 
attain  a  maximum  value.  This  limiting  or  saturation  current 
affords  the  most  accurate  method  for  the  measurement  of  radio- 
activity. 

The  method  is  so  sensitive  that,  by  means  of  it  alone,  it  is  possible 
to  detect  amounts  of  radioactive  products  far  beyond  the  reach  of 
the  balance  or  the  spectroscope.  The  theory  of  gaseous  ionization 
has  been  confirmed  in  several  different  ways,  but  one  of  the  most 
striking  verifications  of  the  theory  is  that  due  to  C.  T.  R.  Wilson. 
Making  use  of  the  fact  that  the  ions  tend  to  condense  water  vapor 
around  themselves  as  nuclei,  Wilson  succeeded  in  actually  photo- 
graphing the  path  of  an  ionizing  ray  in  air. 

Photographic  Action  of  Radiations.  It  has  already  been 
pointed  out  that  the  radiations  from  radioactive  substances  are 
capable  of  affecting  a  photographic  plate.  The  photographic 


46  THEORETICAL  CHEMISTRY 

action  of  the  radiations  has  been  employed  quite  extensively  in 
studying  radioactive  phenomena  from  a  purely  qualitative  stand- 
point. The  method  employed,  consists  in  exposing  the  photo- 
graphic plate,  which  has  been  previously  wrapped  in  opaque  black 
paper,  to  the  action  of  the  radiations.  The  time  of  exposure 
varies  with  the  nature  of  the  substance  under  examination,  a  few 
minutes  being  required  for  highly  active  preparations  while  sev- 
eral days  or  even  weeks  may  be  needed  for  preparations  of  low 
activity. 

Phosphorescence  Induced  by  Radiations.  A  screen  covered 
with  crystals  of  phosphorescent  zinc  sulphide  is  rendered  luminous 
when  exposed  to  fairly  intense  radiation  from  a  radioactive 
substance.  This  phenomenon  has  been  shown  to  be  due  to  the 
bombardment  of  the  crystals  of  zinc  sulphide  by  the  so-called  ex- 
rays  (see  below).  When  the  screen  is  examined  with  a  lens  the 
phosphorescence  is  seen  to  consist  of  a  series  of  scintillations  of 
very  short  duration. 

Nature  of  the  Radiations.  The  ionizing,  photographic,  and 
luminescent  properties  of  the  radiations  from  radioactive  sub- 
stances are  not  sufficient  to  differentiate  them  from  cathode  rays 
or  X-rays,  although  each  of  these  properties  may  be  employed  in 
determining  their  intensity. 

Evidence  of  the  composite  character  of  the  radiations  was  fur- 
nished by  a  study  of  their  penetrating  power  as  well  as  by  investi- 
gations of  the  behavior  of  the  rays  when  subjected  to  the  action  of 
magnetic  and  electric  fields. 

A  thin  sheet  of  aluminum  or  a  few  centimeters  of  air  was  found 
sufficient  to  cut  off  a  large  percentage  of  the  rays. 

The  unabsorbed  portion  of  the  radiation  was  found  to  consist  of 
two  distinct  types,  one  of  which  was  cut  off  by  five  or  six  millimeters 
of  lead  while  the  other  possessed  such  great  penetrating  power 
that  its  presence  could  be  readily  detected  after  passing  through  a 
layer  of  lead  fifteen  centimeters  thick. 

Rutherford  named  these  three  distinct  types  of  radiation,  the 
a-,  /3-,  and  7-rays,  respectively.  The  penetrating  powers  of  the 
a-,  jS-,  and  7-rays  may  be  approximately  expressed  by  the  propor- 
tion 1  to  100  to  10,000;  i.e.,  the  0-rays  are  100  times  more  pene- 


RADIOACTIVITY  47 

trating  than  the  a-rays,  while  the  7-rays  are  100  times  more 
penetrating  than  the  /3-rays. 

The  general  characteristics  of  the  three  kinds  of  rays  may  be 
briefly  summarized  as  follows :  — 

(1)  a-Rays.     The  oi-rays  consist  of  positively  charged  particles 
moving  with  speeds  approximately  one-tenth  as  great  as  that 
of  light.     These  particles  have  been  shown  to  be  identical  with 
helium  atoms  carrying  two  positive  charges  of  electricity.     They 
are  appreciably  deflected  from  a  rectilinear  path  by  magnetic  and 
electric  fields.     They  possess  great  ionizing  power  but  relatively 
little  penetrating  power  or  photographic  action.     The  depth  to 
which  an  a-particle  penetrates  a  homogeneous  absorbing  medium 
before  losing  its  ionizing  power,  is  known  as  its  "  range."     The 
range  is  proportional  to  the  cube  of  the  initial  speed  of  the  a- 
particle  and  is  one  of  the  characteristic  properties  of  the  radio- 
elements  emitting  a-rays. 

(2)  (3-Rays.     The  0-rays  consist  of  negatively  charged  particles 
moving  with  speeds  varying  from  two-fifths  to  nine-tenths  of  the 
speed  of  light.     They  are,  in  fact,  electrons  moving  with  much 
greater  speeds  than  those  shot  out  from  the  cathode  in  a  vacuum 
tube.     While  the  a-particles  emitted  by  a  particular  radio-element 
have  a  definite  velocity,  the  corresponding  /3-ray  emission  consists 
of  a  flight  of  particles  having  widely  different  speeds.     The  pene- 
trating power  of  the  0-rays  is  conditioned  by  the  speed  of  the 
particles,  those  which  move  most  rapidly  possessing  the  greatest 
penetrating  power. 

The  ionizing  action  of  the  /3-rays  is  much  weaker  than  that  of  the 
a-rays,  while  exactly  the  reverse  is  true  of  the  photographic  action. 

(3)  j-Rays.     The  7-rays  are  identical  with  X-rays.     They  con- 
sist of  extremely  short  waves  of  light,  the  wave-length  varying 
from  about  1  X  10~8  cm.  for  the  rays  of  low  penetrating  power  to 
about  1  X  10~9  cm.  for  the  most  penetrating  rays.     Obviously  the 
7-rays  cannot  be  deflected  from  a  rectilinear  path  by  either  electric 
or  magnetic  fields. 

Uranium-X  and  Thorium-X.     In  1900,  Crookes  *  precipitated 
a  solution  of  a  uranium  salt  with  ammonium  carbonate;  when  an 
*  Proc.  Roy.  Soc.,  64,  409  (1900). 


48  THEORETICAL  CHEMISTRY 

excess  of  the  reagent  had  been  added,  all  but  a  minute  portion  of 
the  precipitate  was  found  to  have  dissolved.  This  small  insoluble 
residue,  though  chemically  free  from  uranium,  was  found,  when 
tested  photographically,  to  be  several  hundred  times  more  active, 
weight  for  weight,  than  the  original  salt.  The  solution,  on  the 
other  hand,  was  found  to  have  lost  nearly  all  of  its  activity.  At 
the  end  of  a  year,  however,  the  solution  had  entirely  regained  its 
original  activity,  while  the  insoluble  residue  had  become  inactive. 
The  active  substance  thus  separated  was  called,  on  account  of  its 
unknown  nature,  uranium-X. 

Similarly,  Rutherford  and  Soddy,*  by  precipitating  a  solution 
of  a  thorium  salt  with  ammonium  hydroxide,  found  that  a  large 
proportion  of  the  activity  remained  behind  in  the  thorium-free 
filtrate.  On  evaporating  the  filtrate  to  dryness  and  driving  off 
the  ammonium  salts,  a  residue  was  obtained  which  was,  weight 
for  weight,  several  thousand  times  more  active  than  the  original 
solution.  After  standing  for  a  month,  this  residue  was  found  to 
have  lost  its  activity,  while  the  precipitate  had  regained  the  ac- 
tivity of  the  original  thorium  compound.  This  active  residue  was 
called  thorium-X  from  analogy  to  Crookes'  uranium-X. 

The  fact  that  uranium-X  and  thorium-X  had  each  been  obtained 
as  the  result  of  specific  chemical  processes,  seemed  to  warrant  the 
conclusion  that  they  are  new  substances  possessing  well-defined 
properties.  The  manner  in  which  these  substances  were  obtained 
led  to  a  variety  of  speculations  as  to  the  mechanism  of  the  process 
involved  in  their  production.  In  a  subsequent  paragraph  it  will 
be  shown  that  the  so-called  disintegration  theory  offers  a  most 
satisfactory  explanation  of  the  foregoing  experimental  results. 

The  Emanations.  The  element  thorium  was  found  by  Ruther- 
ford to  give  off  a  radioactive  gas  or  emanation  which,  when  left  to 
itself,  rapidly  loses  its  activity  in  a  similar  manner  to  uranium-X 
and  thorium-X.  The  thorium  emanation  was  found  to  resemble 
the  inactive  gases  in  its  chemical  behavior.  Thus,  it  can  be  sub- 
jected to  the  action  of  lead  chromate,  metallic  magnesium  and 
zinc  dust  at  extremely  high  temperatures  without  undergoing 
change.  The  only  other  substances  which,  at  the  time  of  the  dis- 
*  Phil.  Mag.,  VI,  4,  370  (1902). 


RADIOACTIVITY  49 

covery  of  the  emanation,  were  known  to  resist  the  action  of  these 
reagents  under  the  same  conditions  were  the  gaseous  elements 
helium,  neon,  argon,  krypton,  and  xenon.  The  most  conclusive 
evidence  of  the  gaseous  character  of  the  thorium  emanation  is  the 
fact  that  it  condenses  to  a  liquid  at  very  low  temperatures. 

Rutherford  and  Soddy  showed  that  the  origin  of  the  thorium 
emanation  is  thoriurn-X  and  not  the  element  thorium  itself. 
Freshly  precipitated  thorium  hydroxide  shows  only  a  trace  of 
emanating  power,  whereas  the  thorium-X  separated  from  the 
filtrate  possesses  this  power  to  a  marked  degree.  As  the  thorium- 
X  gradually  loses  its  emanating  power,  the  hydroxide  shows  a 
corresponding  recovery. 

Radium  was  found  to  give  off  an  emanation  which  behaves 
similarly  to  the  thorium  emanation  except  that  it  parts  with  its 
activity  at  a  slower  rate.  The  rate  at  which  the  radium  emanation 
is  produced,  together  with  its  longer  life,  enabled  Ramsay  and 
Soddy  *  not  only  to  measure  the  volume  of  the  emanation  obtained 
from  60  mg.  of  radium  bromide  but  also  to  establish  the  fact  that 
it  obeys  Boyle's  law. 

The  Active  Deposits.  It  was  discovered  by  Rutherford  f  for 
thorium,  and  by  M.  and  Mme.  Curie  t  for  radium,  that  the  emana- 
tions from  these  elements  are  capable  of  imparting  radioactivity 
to  surrounding  objects.  On  the  other  hand,  uranium  and  polo- 
nium, which  evolve  no  emanations,  have  no  such  influence  on  their 
environment.  This  fact  is  taken  as  a  proof  that  induced  radio- 
activity is  due  to  actual  contact  with  the  emanations.  If  a  nega- 
tively charged  platinum  wire  be  exposed  to  the  thorium  or  radium 
emanation,  the  whole  of  its  activity  will  be  concentrated  on  the 
wire.  This  so-called  active  deposit  may  be  transferred  from  the 
wire  to  a  piece  of  sandpaper  by  rubbing.  It  may  be  sublimed 
from  the  wire  to  the  walls  of  the  tube  in  which  it  is  heated,  or  it 
may  be  dissolved  from  the  wire  by  means  of  hydrochloric  or  sul- 
phuric acid.  On  evaporating  the  solution,  the  activity  will  be  found 
to  reside  on  the  evaporating  dish.  Microscopic  examination  of 

*  Proc.  Roy.  Soc.,  73,  346  (1904). 
f  Phil.  Mag..  VI,  22,  621  (1911). 
%  Compt.  rendus,  129,  714  (1899). 


50  THEORETICAL  CHEMISTRY 

the  wire  fails  to  reveal  any  deposit,  and  the  most  sensitive  balance 
is  incapable  of  detecting  any  gain  in  weight  after  exposing  the  wire 
to  the  emanation.  These  experiments  leave  no  room  for  doubt 
that  the  active  deposits  consist  of  infinitesimal  amounts  of  solid 
substances  possessing  definite  chemical  properties.  The  active 
deposits  undergo  a  gradual  loss  of  activity  similar  to  that  observed 
with  the  emanations  and  with  uranium-X  and  thorium-X. 

The  Disintegration  Hypothesis.  It  was  soon  discovered  that 
the  active  deposits  undergo  a  series  of  additional  radioactive 
changes.  These  subsequent  transformations  were  found  to  be 
much  more  obscure  and  difficult  to  follow.  In  fact,  it  is  only  be- 
cause of  the  ingenuity  and  mathematical  acumen  of  those  who 
undertook  this  difficult  research  that  we  are  today  in  possession  of 
such  complete  knowledge  of  the  succeeding  members  of  the  radio- 
active series  of  elements. 

In  1003,  Rutherford  and  Soddy  *  brought  forward  their  theory 
of  atomic  disintegration  which  affords  a  perfectly  satisfactory  in- 
terpretation of  the  complicated  results  already  detailed,  as  well  as 
of  the  subsequent  changes  in  the  active  deposits  to  which  reference 
has  just  been  made.  According  to  this  theory,  radioactive  change 
is  assumed  to  be  due  to  the  spontaneous  disintegration  of  the  radio- 
elements  with  concomitant  formation  of  new  elements.  These 
new  elements,  which  are  often  less  stable  than  the  parent  element, 
are  assumed  to  undergo  further  disintegration  with  the  production 
of  still  other  elements,  the  end  of  the  process  ultimately  being 
reached  when  a  stable  element  is  formed. 

The  Radioactive  Constant.  The  activity  of  a  radio-element 
decays  exponentially  with  time  according  to  the  equation  — 

It  =  /oe-",  (1) 

where  /0  is  the  initial  activity,  It  the  activity  at  the  end  of  an  in- 
terval of  time  t,  X  a  constant,  and  e  is  the  base  of  the  natural  system 
of  logarithms.  The  constant  X,  known  as  the  radioactive  constant, 
represents  the  fraction  of  the  total  amount  of  radioactive  substance 
undergoing  disintegration  in  unit  of  time,  provided  the  latter  is  so 
small  that  the  quantity  at  the  end  of  the  time  unit  is  only  slightly 
*  Phil.  Mag.,  VI,  5,  576  (1903). 


RADIOACTIVITY  51 

different  from  the  initial  quantity.  The  reciprocal  of  the  radio- 
active constant  is  called  the  average  life  of  the  element.  Soddy 
defines  the  average  life  of  a  radio-element  as  "the  sum  of  the  sepa- 
rate periods  of  future  existence  of  all  the  individual  atoms  divided 
by  the  number  in  existence  at  the  starting  point." 

If  nt  represents  the  number  of  atoms  of  a  radio-element  changing 
in  unit  time  at  the  end  of  a  time  t,  and  n0  the  corresponding  value 
when  t  =  0,  equation  (1)  may  be  written 

nt  =  noe~xt. 

In  order  to  determine  the  initial  rate  of  change,  let  No  denote  the 
total  number  of  atoms  originally  present,  and  Nt  the  number  re- 
maining unchanged  at  time  t;  we  then  have 

^0  -v  / 
£> — Al 


But  when  t  =  0,  N0  =  Nt 

and  No  =  - 

A 

Hence  Nt  = 

On  differentiating,  we  have 


Or,  stated  in  words,  the  rate  at  which  the  atoms  of  a  radio-element 
undergo  disintegration  at  any  given  time  is  found  to  be  propor- 
tional to  the  total  number  in  existence  at  that  time. 

This  law  of  radioactive  change  is  also  peculiar  to  unimolecular 
chemical  reactions  (see  p.  361).  The  velocity  of  a  unimolecular 
reaction,  however,  is  conditioned  by  the  temperature,  whereas 
the  velocity  of  a  radioactive  change  remains  unaltered  at  the 
highest  and  the  lowest  attainable  temperatures. 

The  time  required  for  one-half  of  a  radio-element  to  undergo 
transformation  is  known  as  the  period  of  half  change  T,  and  may  be 
readily  calculated  from  X  as  follows :  — 

log  0.5  =  0.4343  \T 

T  = 


52  THEORETICAL  CHEMISTRY 

It  has  been  shown  by  Geiger  and  Nuttall  *  that  the  radioactive 
constant  X  and  the  range  R  of  the  ^-particles  shot  out  from  a  dis- 
integrating atom  bear  the  following  empirical  relation  to  each 
other, 

X  =  aRb, 

where  a  and  b  are  constants,  the  former  constant  being  character- 
istic of  the  particular  radioactive  series  to  which  the  element  be- 
longs. This  formula  has  been  found  useful  in  calculating  the 
values  of  X  for  the  longest  and  shortest  lived  elements. 

Radioactive  Equilibrium.  It  is  evident  that  a  state  of  equi- 
librium must  ultimately  be  attained  among  the  atoms  of  a  radio- 
active substance.  When  the  rate  of  production  of  a  radio-element 
from  its  parent  element  is  equal  to  its  rate  of  disintegration  into 
the  next  succeeding  element  of  the  series,  the  substance  is  said  to 
be  in  radioactive  equilibrium.  The  relative  amounts  of  the  suc- 
cessive members  of  a  series  of  elements  in  radioactive  equilibrium 
are  inversely  proportional  to  their  radioactive  constants. 

In  order  that  measurements  of  the  rate  of  radioactive  change 
may  be  strictly  comparable,  it  is  necessary  to  make  use  of  the 
amounts  which  are  in  equilibrium  with  a  fixed  amount  of  the 
parent  element.  Thus,  the  unit  adopted  for  the  measurement  of 
the  quantity  of  the  radium  emanation  is  the  mass  of  emanation  in 
equilibrium  with  one  gram  of  radium.  This  unit  is  known  as  the 
curie.  Its  mass  is  Xi/X2  gram,  where  Xi  and  X2  are  the  radioactive 
constants  of  radium  and  its  emanation  respectively.  One  curie 
of  radium  emanation  may  be  shown  to  occupy  0.63  cu.  mm.  under 
standard  conditions  of  temperature  and  pressure. 

The  Disintegration  Series.  It  appears  almost  certain  that  the 
thirty  or  more  radio-elements  are  disintegration  products  of  one  or 
the  other  of  the  two  parent  elements,  uranium  or  thorium.  One 
of  the  most  convenient  methods  of  classification  of  these  elements 
is  to  arrange  them  in  disintegration  series,  starting  with  the  parent 
element  and  placing  the  succeeding  elements  in  the  order  of  their 
production.  The  accompanying  table  shows  the  three  series  of 
radio-elements  as  thus  arranged  by  Soddy.  The  numbers  within 
the  circles  are  the  atomic  weights  of  the  elements,  while  the  small 
*  Phil.  Mag.,  VI,  22,  613  (1911). 


RADIOACTIVITY 
TABLE  SHOWING  DISINTEGRATION  SERIES 


53 


54  THEORETICAL  CHEMISTRY 

circles  and  dots  at  the  right  of  the  larger  circles  indicate  the  char- 
acter of  the  radiation  given  out  at  each  stage  of  the  disintegration 
process.  The  average  life,  1/X,  of  each  element  in  the  series  is 
given  below  the  name  of  the  element.  While  a  detailed  account 
of  the  properties  of  the  different  radio-elements  included  in  this 
table  cannot  be  given  here,  attention  should  be  called  to  the  com- 
plex transformations  occurring  in  the  active  deposit.  It  is  also  of 
interest  to  observe  that  the  end-products  of  the  three  series  bear  a 
striking  resemblance  to  the  element  lead  and  that  their  atomic 
weights  are  approximately  equal  and  nearly  identical  with  that 
of  ordinary  lead. 

Counting  the  a-Particles.  It  has  already  been  stated  that 
when  an  a-particle  strikes  a  screen  coated  with  phosphorescent 
zinc  sulphide,  a  distinct  flash  of  light  may  be  seen  when  the  screen 
is  viewed  through  a  magnifying  lens  in  a  dark  room.  It  is  obvious 
that  if  one  could  count  the  number  of  these  scintillations,  it  would 
be  an  easy  matter  to  ascertain  the  total  number  of  a-particles 
shot  out  from  a  radioactive  substance  in  a  given  time.  By  using 
a  phosphorescent  screen  and  a  microscope,  Regener*  has  deter- 
mined in  this  manner  the  rate  of  emission  of  a-particles  from 
polonium.  He  found  from  his  different  experiments  an  average 
emission  of  3.94  X  105  a-particles  per  second.  The  total  charge 
on  the  a-particles  was  then  measured  by  collecting  them  in  a  suit- 
able measuring  vessel.  A  charge  of  37.7  X  10~5  electrostatic 
units  was  found  to  be  associated  with  the  total  number  of  a- 
particles  emitted  by  polonium  in  one  second. 

Rutherford  and  Geiger  f  devised  an  electrical  method  for  count- 
ing the  a-particles.  In  their  experiment  the  source  of  the  a- 
particles  was  a  small  disc  which  had  been  exposed  to  the  radium 
emanation  for  some  hours.  This  disc  was  placed  in  an  evacuated 
tube  at  a  measured  distance  from  a  small  aperture  of  known  cross- 
section.  The  aperture  was  closed  with  a  thin  plate  of  mica  through 
which  the  a-particles  could  pass  with  ease.  After  passing  through 
the  mica  plate,  the  a-particles  entered  an  ionization  chamber 
filled  with  air  at  reduced  pressure  and  fitted  with  two  charged 

*  Sitzunsbericht  d.  K.  preuss.  Akad.,  38,  948  (1909). 
t  Proc.  Roy.  Soc.  A,  81,  141  (1908). 


RADIOACTIVITY  55 

metal  plates  connected  with  appropriate  apparatus  for  measuring 
ionization  currents.  Whenever  an  a-particle  entered  the  ioniza- 
tion  chamber,  a  momentary  current  passed,  producing  a  sudden 
deflection  of  the  needle  of  the  electrometer.  By  counting  the 
number  of  throws  of  the  needle  occurring  in  a  definite  interval  of 
time,  the  total  number  of  a-particles  passing  through  the  ionization 
chamber  was  determined.  Knowing  the  distance  of  the  source 
of  the  radiations  from  the  aperture,  together  with  the  area  of  the 
aperture,  the  total  number  of  a-particles  emitted  by  the  radio- 
active disc  in  a  given  time  could  be  computed.  Rutherford  and 
Geiger  thus  found  that  one  gram  of  radium  emits  very  nearly 
107  X  1016  a-particles  per  year.  Having  determined  the  total 
number  of  a-particles  emitted,  it  only  remained  to  measure  the 
total  charge,  in  order  to  calculate  the  charge  carried  by  a  single 
a-particle.  From  a  series  of  very  consistent  measurements,  the 
charge  carried  by  a  single  a-particle  was  found  to  be  9.3  X  10~10 
electrostatic  units.  Since  the  fundamental  charge  e  has  been 
shown  to  be  4.48  X  10~10  electrostatic  units,  it  follows  that  the 
o:-particle  carries  two  ionic  charges  of  electricity. 

Helium  Atoms  and  a-Particles  Identical.  In  1909,  Ruther- 
ford and  Royds  *  performed  a  crucial  experiment  to  determine  the 
nature  of  the  a-particle.  A  glass  bulb  was  blown  with  walls  thin 
enough  to  permit  the  passage  of  the  a-particles  but  sufficiently 
strong  to  withstand  atmospheric  pressure.  The  bulb  was  filled 
with  radium  emanation  and  then  enclosed  in  an  outer  glass  tube 
to  which  a  spectrum  tube  had  been  sealed.  On  exhausting  the 
outer  tube  and  examining  the  spectrum  of  the  residual  gas,  no 
evidence  of  helium  was  obtained  until  after  an  interval  of  twenty- 
four  hours.  After  four  days  the  characteristic  yellow  and  green 
lines  were  plainly  visible  and  at  the  end  of  the  sixth  day,  the  com- 
plete spectrum  of  helium  was  obtained.  The  unavoidable  con- 
clusion from  this  experiment  is,  that  the  presence  of  helium  in  the 
outer  tube  must  have  been  due  to  the  a-particles  which  were 
projected  through  the  thin  walls  of  the  inner  tube.  In  another 
experiment,  the  inner  tube  was  filled  with  pure  helium  under 
pressure  while  the  exhausted  outer  tube  was  examined  for  helium. 
*  Phil.  Mag.,  VI,  17,  281  (1909). 


56  THEORETICAL  CHEMISTRY 

No  trace  of  helium  could  be  detected  spectroscopically  even  after 
an  interval  of  several  days,  thus  proving  that  the  helium  detected 
in  the  first  experiment  must  have  resulted  from  the  a-particles 
which  had  been  shot  out  from  the  radium  emanation  with  sufficient 
energy  to  penetrate  the  thin  walls  of  the  inner  tube.  These  experi- 
ments leave  no  room  for  doubt  that  an  a-particle  becomes  a 
helium  atom  when  its  positive  charge  is  neutralized. 

Rate  of  Production  of  Helium.  The  rate  of  production  of 
helium  from  the  series  in  equilibrium  with  one  gram  of  radium 
has  been  determined  experimentally  by  Rutherford  and  Bolt- 
wood*  to  be  156  cu.  mm.  per  year.  This  result  agrees  closely 
with  the  calculated  rate  of  production,  viz.,  158  cu.  mm.  per  gram 
of  radium  per  year. 

Energy  Evolved  by  Radium.  Curie  and  Laborde  f  were  the 
first  to  call  attention  to  the  interesting  fact  that  the  temperature 
of  radium  compounds  was  uniformly  higher  than  that  of  their 
environment.  Careful  measurements  have  shown  that  one  gram 
of  radium  evolves  heat  at  the  rate  of  approximately  135  gram- 
calories  per  hour.  That  the  greater  part  of  this  heat  energy  is  due 
to  the  a-particles  may  be  proven  by  a  direct  calculation  of  their 
mean  kinetic  energy.  The  magnitude  of  the  store  of  energy  con- 
tained in  radium  may  be  realized  from  the  statement,  that  one 
gram  of  radium,  before  it  entirely  disappears,  evolves  an  amount 
of  heat  energy  nearly  one  million  times  greater  than  that  evolved 
in  the  formation  of  one  gram  of  water  from  its  elements. 
*  Phil.  Mag.,  VI,  22,  586  (1911). 
t  Compt.  rendus,  136,  673  (1904). 


CHAPTER  V. 
ATOMIC   STRUCTURE. 

The  Modern  Conception  of  Atomic  Structure.  As  a  result  of 
the  investigations  of  Thomson,*  Rutherford,f  Nicholson,!  Bohr,§ 
and  others,  a  theory  of  atomic  structure  has  been  developed  which 
affords  a  satisfactory  interpretation  of  many  of  the  important 
relationships  among  the  chemical  elements. 

Briefly  stated,  this  theory  assumes  that  the  atom  consists  of  a 
central,  positively  charged  nucleus,  surrounded  by  a  miniature 
solar  system  of  electrons.  The  investigations  of  Rutherford  and 
Geiger||  show  that  the  character  of  the  deflection  of  a-particles 
shot  out  from  radioactive  atoms  at  speeds  approximating  20,000 
miles  per  second,  and  consequently  completely  penetrating  other 
atoms,  is  such  as  to  indicate  an  extremely  high  concentration  of 
positive  electricity  on  the  central  nucleus.  The  central  nucleus 
which  is  supposed  to  represent  nearly  the  entire  mass  of  the  atom, 
is  thought  to  be  very  small  in  comparison  with  the  size  of  the  atom 
as  a  whole.  Recent  investigations  make  it  appear  probable  that 
the  maximum  diameter  of  the  nucleus  of  the  hydrogen  atom  is 
about  one  one-hundred-thousandth  of  the  diameter  commonly 
attributed  to  the  atom.  In  commenting  on  this  statement,  Har- 
kins  says:  1f  —  "On  this  basis  the  atom  would  have  a  volume  a 
million-billion  times  larger  than  that  of  its  nucleus,  and  thus  the 
nucleus  of  the  atom  is  much  smaller  in  comparison  with  the  size 
of  the  atom  than  is  the  sun  when  compared  with  the  dimensions 
of  its  planetary  system."  It  is  highly  probable  that  the  central 
nucleus  is  itself  made  up  of  a  definite  number  of  units  of  positive 
electricity  together  with  a  small  number  of  attendant  electrons. 

*  Phil.  Mag.,  7,  237  (1904). 
t  Popular  Science  Monthly,  87,  105  (1915). 
{  Phil.  Mag.,  22,  864  (1911). 
§  Phil.  Mag.,  26,  476,  857  (1913). 
H  Phil.  Mag,  21,  669  (1911). 
1f  Science,  66,  419  (1917). 
57 


58  THEORETICAL  CHEMISTRY 

It  is  further  assumed  that  the  units  of  positive  electricity  are 
hydrogen  atoms,  each  of  which  has  been  deprived  of  one  electron. 
If  the  mass  of  an  atom  is  largely  due  to  the  presence  of  hydrogen 
nuclei,  then  we  should  expect  Prout's  hypothesis  to  hold  and  the 
atomic  weights  of  the  elements  to  be  exact  multiples  of  the  atomic 
weight  of  hydrogen.  When  we  consider,  however,  that  according 
to  electromagnetic  theory  the  total  mass  of  a  body  composed  of 
positive  and  negative  units  is  dependent  upon  the  relative  posi- 
tions of  these  units  when  packed  together,  it  is  evident  that  the 
mass  of  the  atom  will  not  necessarily  be  an  exact  multiple  of  the 
mass  of  the  hydrogen  atom. 

It  has  already  been  pointed  out  that  helium  is  a  product  of 
many  radioactive  transformations.  This  fact  may  be  taken  as  an 
indication  of  the  extraordinary  stability  of  the  helium  atom.  Be- 
cause of  its  stability,  the  nucleus  of  this  atom  has  come  to  be  con- 
sidered as  a  secondary  unit  of  positive  electricity.  The  nucleus 
of  the  helium  atom,  or  the  nucleus  of  an  ex-particle  is  assumed  to 
consist  of  four  hydrogen  nuclei  with  two  nuclear  electrons. 

The  Atomic  Number.  Since  the  algebraic  sum  of  the  positive 
and  negative  electrification  on  an  atom  must  be  zero,  it  follows 
that  the  charge  resident  upon  the  nucleus  must  be  equal  to  the 
number  of  electrons  outside  the  nucleus.  This  number,  which  has 
come  to  be  recognized  as  more  important  and  characteristic  than 
the  atomic  weight,  is  known  as  the  atomic  number. 

X-Rays  and  Atomic  Structure.  The  discovery  by  W.  L.  Bragg 
in  1912  that  X-rays  undergo  reflection  at  crystal  surfaces  and  the 
subsequent  development  by  Mr.  Bragg  and  his  father,  W.  H.  Bragg, 
of  the  X-rays  pectrometer,  has  led  to  a  series  of  investigations  of 
the  utmost  importance  to  both  the  chemist  and  the  physicist. 

In  order  that  the  significance  of  these  investigations  may  be 
understood  it  may  be  well  to  summarize  very  briefly  some  of  the 
more  important  properties  of  the  X-ray. 

The  bombardment  of  metal  plates,  usually  of  platinum,  by  elec- 
trons gives  rise  to  X-rays.  The  radiation  issuing  from  an  X-ray 
tube  is  very  far  from  homogeneous.  When  screens  of  different 
materials  and  varying  thicknesses  are  interposed  in  the  path  of 
the  rays,  the  degree  of  absorption  is  irregular.  It  has  been  found, 


ATOMIC  STRUCTURE  59 

however,  that  every  substance  when  properly  stimulated  is  capable 
of  emitting  a  homogeneous  and  characteristic  X-radiation,  the 
penetrating  power  of  which  is  wholly  determined  by  the  nature  of 
the  elements  of  which  the  substance  is  composed.  The  pene- 
trating power  of  this  typical  X-radiation  increases  with  the  atomic 
weight  of  the  radiating  element.  With  elements  whose  atomic 
weights  are  less  than  24,  the  radiation  is  too  feeble  to  be  measured. 
It  is  important  to  note  that  this  property  of  the  elements  is  not  a 
periodic  function  of  the  atomic  weight.  This  type  of  X-radiation 
is  entirely  independent  of  external  conditions,  indicating  that  it 
is  closely  connected  with  the  internal  structure  of  the  atoms  from 
which  it  emanates.  The  rays  possess  the  power  of  affecting  the 
photographic  plate  and  also  of  rendering  gases  through  which 
they  pass  conductors  of  electricity. 

It  is  estimated  that  the  wave-length  of  an  X-ray  is  about  1 X  10~8 
to  1  X  10~9  cm.,  or  about  one  ten-thousandth  of  the  wave-length 
of  sodium  light.  It  is  obvious  that  the  spacing  of  the  lines  of  a 
grating  capable  of  diffracting  such  short  waves  must  be  of  the 
order  of  magnitude  of  intermolecular  distances.  It  is  well  known 
that  a  grating  owes  its  power  of  analyzing  a  complex  system  of 
light  waves  into  its  component  wave-trains,  to  the  series  of  paral- 
lel lines  which  are  ruled  upon  its  surface  at  exactly  equal  intervals. 
When  a  train  of  waves  is  incident  upon  a  grating,  each  line  acts 
as  a  center  from  which  a  diffracted  train  of  waves  emerges. 

Such  a  grating  is  relatively  simple  in  its  action  since  it  consists 
of  a  single  series  of  centers  of  diffraction  lying  in  one  plane.  The 
power  of  a  crystal  surface  to  reflect  X-rays,  however,  is  due  to  the 
fact  that  the  crystal  is  in  reality  a  three-dimensional  diffraction 
grating,  the  atoms  or  molecules  of  which  the  crystal  is  built  up, 
acting  as  the  centers  of  diffraction.  It  must  be  borne  in  mind  that 
the  reflection  of  X-rays  is  in  no  way  dependent  upon  the  existence 
of  a  polished  surface  on  the  outside  of  the  crystal,  but  rather  upon 
the  regularly  spaced  atoms  or  molecules  within  the  crystal.  To 
ordinary  waves  of  light  the  atomic  structure  is  so  fine  grained  as 
to  behave  as  a  continuous  medium,  whereas  to  the  short  X-ray 
waves,  the  crystal  acts  as  a  discontinuous  structure  of  regularly 
arranged  particles,  each  of  which  functions  as  a  diffraction  center. 


60 


THEORETICAL  CHEMISTRY 


Spectra.  By  making  use  of  the  reflecting  power  of  one 
of  the  cleavage  planes  of  a  crystal,  and  employing  different  metals 
as  anti-cathodes  in  an  X-ray  tube,  Moseley  *  succeeded  in  photo- 
graphing the  X-ray  spectra  of  the  characteristic  radiations  of  a 
number  of  the  elements.  He  showed  that  the  X-ray  spectrum  of 
an  element  is  extremely  simple  and  consists  of  two  groups  of  lines 
known  as  the  "K  "  and  "L  "  radiations.  As  a  result  of  careful 
study  of  the  "K  "  radiations  of  thirty-nine  elements  from  alumin- 
ium to  gold,  Moseley  discovered  that  these  radiations  are  char- 
acterized by  two  well-defined  lines  whose  vibration  frequency  v 
is  connected  with  the  atomic  number  of  the  element  AT,  by  the 
simple  relation  v  =  A  (N  —  I)2, 

where  A  is  a  constant.     When  the  square  roots  of  the  frequencies 
of  the  elements  are  plotted  as  abscissae  against  their  atomic  num- 
bers as    ordinates, 
the  points  are  found 
to  lie  on  a  straight 

50 1 1 1 1 *j — T^— I  loo      line  as  shown  in  Fig. 

11.  On  the  other 
hand,  if  the  square 
roots  of  the  frequen- 
cies are  plotted 
against  the  atomic 
weights  of  the  ele- 
ments, the  relation- 

20 1 ^4 1 1 1 1  40      ship  is  no  longer  rec- 
tilinear.   When  the 
elements   are   ar- 
ranged in  the  order 
of  their  atomic  num- 
bers instead  of  in  the 
order  of  their  atomic  weights,  the  irregularities  f  hitherto  noted  in 
connection  with  argon,  cobalt,  and  tellurium  entirely  disappear. 
In  reviewing  Moseley's  work  on  X-ray  spectra,  Soddy  {  says :  — 

*  Phil.  Mag.,  26,  210  (1913);  27,  703  (1914). 

t  See  p.  29. 

J  Ann.  Reports  on  the  Prog,  of  Chemistry,  p.  278  (1914). 


80 


10  14  18  22 

Square  Root  of  Frequency  X  10"8 

Fig.  11. 


-ATOMIC  STRUCTURE  61 

"A  veritable  roll-call  of  the  elements  has  been  made  by  this 
method.  Thirty-nine  elements,  with  atomic  weights  between 
those  of  aluminium  and  gold,  have  been  examined  in  this  way,  and 
in  every  case  the  lines  of  the  X-ray  spectrum  have  been  found  to 
be  simply  connected  with  the  integer  that  represents  the  place 
assigned  to  it  by  chemists  in  the  periodic  table." 

One  of  the  most  interesting  results  of  this  "  roll-call  "  of  the 
elements  is  the  fixing  of  the  number  of  possible  rare-earth  elements. 
Between  barium  and  tantalum  there  are  places  for  only  fifteen 
rare-earth  elements  and  fourteen  of  these  places  are  filled.  While 
future  investigation  may  necessitate  some  rearrangement  in  the 
order  of  tabulation,  the  total  number  of  these  elements  is  limited 
to  fifteen. 

Periodicity  among  the  Radio-elements.  The  problem  of  plac- 
ing the  newly  discovered  radio-elements  in  the  periodic  table  re- 
mained unsolved  until  1913,  when  Fajans  *  and  Soddy,f  working 
independently,  discovered  an  important  generalization  concerning 
the  changes  in  chemical  properties  resulting  from  the  expulsion  of 
a-  and  /3-particles  during  radioactive  transformation.  This  impor- 
tant generalization  may  be  stated  as  follows:  —  The  expulsion  of 
an  a-particle  causes  a  radioactive  element  to  shift  its  position  in 
the  periodic  table  two  places  in  the  direction  of  decreasing  atomic 
weight,  whereas  the  emission  of  a  (3-particle  causes  a  shift  of  one 
place  in  the  opposite  direction.  This  generalization  not  only  agrees 
with  our  present  theory  of  atomic  structure,  but  may  be  shown 
to  be  a  necessary  consequence  of  this  theory. 

The  loss  of  an  a-particle  or  helium  atom  involves  a  loss  of  4 
units  in  atomic  weight  and  of  2  units  of  positive  electricity  from 
the  nucleus  of  the  atom.  In  consequence  of  this  loss,  the  atomic 
number  is  diminished  by  2  units  and  the  resulting  new  element 
will  find  a  place  in  the  periodic  table  two  groups  to  the  left  of  that 
occupied  by  the  parent  element.  On  the  contrary,  while  the 
expulsion  of  a  /3-particle,  or  electron,  involves  practically  no  change 

mass,  the  nucleus  of  the  parent  atom  suffers  a  loss  of  1  unit  of 
egative  electricity.  This  loss  is  equivalent  to  a  gain  of  1  unit  of 

*  Physikal.  Zeit.,  14,  49  (1913). 
f  Chem.  News,  107,  97  (1913). 


62 


THEORETICAL  CHEMISTRY 


positive  electricity,  or  to  an  increase  of  1  unit  in  the  atomic  number, 
and  in  consequence,  the  position  of  the  new  element  in  the  periodic 
table  will  be  shifted  one  group  to  the  right  of  that  occupied  by  the 
parent  element. 

Soddy's  arrangement  of  all  of  the  radio-elements  in  accordance 
with  this  generalization  is  shown  in  Fig.  12,  Thus,  starting  with 
the  element  uranium  in  Group  VIA,  we  may  follow  the  successive 


RADIO-ELEMENTS  AND   PERIODIC   LAW 
ALL  ELEMENTS  IN  THE  SAME   PLACE 

IN   THE   PERIODIC  TABLE 
ARE  CHEMICALLY  NON-SEPARABLE 

AND   (PROBABLY) 
'ECTROSCOPICALLY  INDISTINGUISHABLE 


Relative  No.  of  Negative  Electrons 
543  1  0 


240 


Fig.  12. 


steps  in  the  radium  disintegration  series  which  was  discussed  in 
the  preceding  chapter.  The  element  UXi  resulting  from  U  by 
the  loss  of  an  a-particle  is  placed  in  Group  IVA.  This  element  in 
turn  undergoes  a  /3-ray  change  producing  the  element  UX2  which 
is  accordingly  placed  in  Group  VA.  The  element  UII  in  Group 
VIA  is  formed  from  UX2  by  0-ray  disintegration,  while  the  ele- 
ment lo  results  from  the  loss  of  an  a-particle  by  UII  with  a  con- 
sequent shifting  of  two  places  to  the  left  in  the  table.  A  similar 
loss  of  an  a-particle  by  lo  brings  us  to  the  element  Ra  in  Group 


ATOMIC  STRUCTURE  63 

IIA.  In  the  successive  steps  of  this  disintegration  from  U  to  Ra, 
three  a-particles  or  12  units  of  atomic  mass  are  lost,  and  the  atomic 
weight  of  Ra,  as  calculated  from  that  of  U,  agrees  with  the  atomic 
weight  found  by  direct  experiment.  In  like  manner  the  remain- 
ing stages  of  the  disintegration  may  be  followed  to  the  end-product 
in  Group  IVB. 

Isotopes.  Perhaps  the  most  striking  feature  in  the  table  is 
the  occurrence  of  several  different  elements  in  the  same  place,  as 
for  example  in  Group  IVB,  where  in  the  place  occupied  by  the 
element  Pb,  we  also  find  RaB,  RaD,  ThB,  and  AcB,  together 
with  four  other  elements  to  which  no  names  have  been  assigned, 
but  which  are  none  the  less  stable  end-products.  The  individual 
members  of  such  a  group  of  elements  occupying  the  same  place  in 
the  periodic  table,  and  being  in  consequence  chemically  identical, 
are  known  as  isotopes.  Isotopic  elements  have  identical  arc  and 
spark  spectra  and,  except  for  differences  in  atomic  weight,  are 
chemically  indistinguishable. 

Making  use  of  the  fact  that  two  gaseous  elements  having  differ- 
ent atomic  weights  diffuse  at  different  rates,  Thomson  and  Aston 
have  recently  succeeded  in  separating  neon  into  two  isotopes 
having  atomic  weights  20  and  22  respectively.  This  is  the  only 
method  which  has  thus  far  given  promise  of  success  in  effecting 
isotopic  separations. 

It  is  interesting  to  note  in  Soddy's  table  (Fig.  12),  that  "the  ten 
occupied  spaces  (groups)  contain  nearly  forty  distinct  elements, 
whereas  if  chemical  analysis  alone  had  been  available  for  their 
recognition,  only  ten  elements  could  have  been  distinguished." 

The  Hydrogen-Helium  System  of  Atomic  Structure.  A 
generalization  similar  to  that  just  outlined  for  the  radio-elements 
has  been  found  by  Harkins  and  Wilson  *  to  hold  true  for  the  lighter 
elements  which  apparently  do  not  undergo  appreciable  a-ray  dis- 
integration. Beginning  with  helium  and  adding  4  units  of  atomic 
weight  for  each  increase  of  2  units  in  the  atomic  number,  gives  the 
atomic  weights  of  the  elements  in  the  even-numbered  groups  of 
the  periodic  table,  neglecting  small  changes  in  mass  due  to  nuclear 
packing.  This  rule  has  been  found  to  hold  very  closely  for  all  of 
the  elements  having  atomic  weights  below  60. 

*  Proc.  Nat.  Acad.,  Vol.  I,  p.  276  (1915). 


64  THEORETICAL  CHEMISTRY 

The  atomic  weights  of  the  elements  of  the  odd-numbered  groups 
can  be  calculated  by  a  similar  rule,  provided  that  the  atom  of 
lithium,  the  first  member  of  the  odd-numbered  groups,  be  assumed 
to  be  made  up  of  1  hydrogen  and  3  helium  nuclei.  The  following 
table  gives  the  results  as  calculated  by  Harkins  and  Wilson  for  the 
first  three  series  of  the  periodic  table. 

The  so-called  theoretical  atomic  weights  are  calculated  on  the 
basis  H  =  1,  while  the  experimentally  determined  values  are  on 
the  basis  O  =  16  or  H  =  1.0078.  The  remarkably  close  agree- 
ment between  the  two  sets  of  values  is  taken  as  an  indication  that 
the  packing  effect,  resulting  from  the  formation  of  the  elements 
from  hydrogen  nuclei  and  attendant  electrons,  is  very  small. 
This  packing  effect  has  been  estimated  to  involve  a  decrease  in 
atomic  mass  of  about  0.77  per  cent,  and  is  believed  to  be  due 
almost  entirely  to  the  formation  of  the  helium  atom.  The  hydro- 
gen-helium hypothesis  of  atomic  structure  offers  a  rational  ex- 
planation of  many  interesting  but  hitherto  obscure  facts  concern- 
ing the  nature  of  the  elements. 

Relation  between  Atomic  Weights  and  Atomic  Numbers.  For 
all  elements  whose  atomic  weights  are  less  than  that  of  nickel, 
Harkins  finds  the  following  simple  mathematical  relation  to  hold, 

w  =  2# +  4  + J(-i)"-1, 

where  W  is  the  atomic  weight  and  N  is  the  atomic  number.  In 
other  words,  the  atomic  weights  are  a  linear  function  of  the  atomic 
numbers. 

The  Periodic  Law.  In  the  light  of  recent  discoveries  the 
periodic  law  acquires  new  significance;  in  fact  to-day  the  periodic 
law  may  be  regarded  as  the  most  comprehensive  generalization  in 
the  whole  science  of  chemistry. 

Attention  has  already  been  directed  in  an  earlier  chapter  to  the 
most  apparent  of  the  imperfections  in  Mendeleeff's  system  of 
classification  of  the  elements.  While  the  later  tables  are  more 
complete  than  the  original,  owing  in  part  to  the  discovery  of  new 
elements,  it  must  be  admitted  nevertheless  that  relatively  little 
real  progress  has  been  made  until  recently  toward  removing  the 
seemingly  inherent  defects  of  the  system. 


ATOMIC  STRUCTURE 


65 


02 

O 

! 

^ 
o 

I 

fe 

w 

I 


+88 

Cirv.    ' 

•^    o  O5  Oi 


8t^ 
0 

c^d 

co  co 


ojoco 
W 


Wo^ 


SI 


a 


bo 

' 


ffi 

+ 

0 
VjH  10  »0 


3  lea 


<B   o 

02    a 


66  THEORETICAL  CHEMISTRY 

A  satisfactory  periodic  table  should  meet  the  following  require- 
ments :  — 

(1)  It  should  afford  a  place  for  isotopic  elements  such  as  lead. 

(2)  The  radio-elements  together  with  their  a-  and  /3-disinte- 
gration  products  should  be  shown. 

(3)  It  should  contain  no  vacant  spaces  except  those  correspond- 
ing to  the  atomic  numbers  of  undiscovered  elements. 

(4)  It  should  bring  out  the  relation  between  the  elements  con- 
stituting a  main  group  and  those  forming  the  corresponding  sub- 
group.    For  example,  the  relation  between  the  elements  Be,  Mg, 
Ca,  Sr,  Ba,  and  Ra  on  the  one  hand,  and  the  elements  Zn,  Cd,  and 
Hg  on  the  other,  should  be  emphasized. 

(5)  The  elements  of  Group  O  and  Group  VIII  should  fit  natu- 
rally in  the  table. 

(6)  All  of  the  foregoing  conditions  should  be  shown  by  means  of 
a  continuous  curve  connecting  the  elements  in  the  order  of  their 
atomic  numbers,  the  latter  having  been  shown  to  be  more  charac- 
teristic of  an  element  than  its  atomic  weight. 

A  table  which  satisfactorily  meets  these  requirements  has  re- 
cently been  devised  by  Harkins  and  Hall.  This  table  may  be 
constructed  in  the  form  of  a  helix  in  space  or  as  a  spiral  in  a  plane. 
The  following  description  of  the  helical  arrangement,  shown  in 
Fig.  13,  is  taken  verbatim  from  the  original  paper  of  Harkins  and 
Hall.* 

"The  atomic  weights  are  plotted  from  the  top  down,  one  unit  of 
atomic  weight  being  represented  by  one  centimeter,  so  the  model 
is  about  two  and  one-half  meters  high.  .  .  . 

"The  balls  representing  the  elements  are  supposed  to  be  strung 
on  vertical  rods.  All  of  the  elements  on  one  vertical  rod  belong  to 
one  group,  have  on  the  whole  the  same  maximum  valence,  and  are 
represented  by  the  same  color.  The  group  numbers  are  given  at 
the  bottom  of  the  rods.  On  the  outer  cylinder  the  electro-nega- 
tive elements  are  represented  by  black  circles  at  the  back  of  the 
cylinder,  and  electro-positive  elements  by  white  circles  on  the  front 
of  the  cylinder.  The  transition  elements  of  the  zero  and  fourth 
groups  are  represented  by  circles  which  are  half  black  and  half 
*  Jour.  Am.  Chem.  Soc.,  38,  169  (1916). 


Or 


80 


100 


140 


/160 


180 


200 


Group 


Fig.  13 


68  THEORETICAL  CHEMISTRY 

white.  The  inner  loop  elements  are  intermediate  in  their  proper- 
ties. Elements  on  the  back  of  the  inner  loop  are  shown  as  heavily 
shaded  circles,  while  those  on  the  front  are  shaded  only  slightly. 
"In  order  to  understand  the  table  it  may  be  well  to  take  an 
imaginary  journey  down  the  helix,  beginning  at  the  top.  Hydro- 
gen (atomic  number  and  atomic  weight  =  1)  stands  by  itself,  and 
is  followed  by  the  first  inert,  zero  group,  and  zero  valent  element 
helium.  Here  there  comes  the  extremely  sharp  break  in  chemical 
properties  with  the  change  to  the  strongly  positive,  univalent 
element  lithium,  followed  by  the  somewhat  less  positive  bivalent 
element,  beryllium,  and  the  third  group  element  boron,  with  a 
positive  valence  of  three,  and  a  weaker  negative  valence.  At  the 
extreme  right  of  the  outer  cylinder  is  carbon,  the  fourth  group 
transition  element,  with  a  positive-  valence  of  four,  and  an  equal 
negative  valence,  both  of  approximately  equal  strength.  The 
first  element  on  the  back  of  the  cylinder  is  more  negative  than 
positive,  and  has  a  positive  valence  of  five,  and  a  negative  valence 
of  three.  The  negative  properties  increase  until  fluorine  is  reached 
and  then  there  is  a  sharp  break  of  properties,  with  the  change  from 
the  strongly  negative,  univalent  element  fluorine,  through  the  zero 
valent  transition  element  neon,  to  the  strongly  positive  sodium. 
Thus  in  order  around  the  outer  loop  the  second  series  of  elements 
are  as  follows :  — 

Group  number 01234567 

Maximum  valence ...0          1          2          3          4          5          6          7 

Element He       Li        Be         B         C         N         0        F 

Atomic  number 2          3          4          5          6          7          8          9 

"  After  these  comes  neon,  which  is  like  helium,  sodium  which  is 
like  lithium,  etc.,  to  chlorine,  the  eighth  element  of  the  second 
period.  For  the  third  period  the  journey  is  continued,  still  on 
the  outer  loop,  with  argon,  potassium,  calcium,  scandium,  and 
then  begins  with  titanium,  to  turn  for  the  first  time  into  the  inner 
loop.  Vanadium,  chromium,  and  manganese,  which  comes  next, 
are  on  the  inner  loop,  and  thus  belong,  not  to  main  but  to  sub- 
groups. This  is  the  first  appearance  in  the  system  of  sub-group 
ekments.  Just  beyond  manganese  a  catastrophe  of  some  sort 
seems  to  take  place,  for  here  three  elements  of  one  kind,  and  there- 


ATOMIC  STRUCTURE  69 

fore  belonging  to  one  group,  are  deposited.  The  eighth  group  in 
this  table  takes  the  place  on  the  inner  loop  which  the  rare  gases  of 
the  atmosphere  fill  on  the  outer  loop.  The  eighth  group  is  thus  a 
sub-group  of  the  zero  group. 

"After  the  eighth  group  elements,  which  have  appeared  for  the 
first  time,  come  copper,  zinc,  and  gallium;  and  with  germanium,  a 
fourth  group  element,  the  helix  returns  to  the  outer  loop.  It  then 
passes  through  arsenic,  selenium,  and  bromine,  thus  completing 
the  first  long  period  of  18  elements.  Following  this  there  comes  a 
second  long  period,  exactly  similar,  and  also  containing  18  elements. 

"The  relations  which  exist  may  be  shown  by  the  following 
natural  classification  of  the  elements.  They  may  be  divided  into 
cycles  and  periods  as  follows: 

TABLE  I. 
Cycle  1=42  elements. 

1st  short  period He  —  F    =    8  =  2X22  elements. 

2nd  short  period Ne  —  Cl  =    8  =  2X22  elements. 

Cycle  2  =  62  elements. 

1st  long  period A  —  Br  =  18  =  2  X  32  elements. 

2nd  long  period Kr  —  I  =  18  =  2  X  32  elements. 

Cycle  3  =  82  elements. 

1st  very  long  period.  ...    Xe  —  Eka-I  =  32  =  2  X  42  elements. 
2nd  very  long  period ....    Nt  —  U 

"The  last  very  long  period,  and  therefore  the  last  cycle,  is  in- 
complete. It  will  be  seen,  however,  that  these  remarkable  relations 
are  perfect  in  their  regularity.  These  are  the  relations,  too,  which 
exist  in  the  completed  system,*  and  are  not  like  many  false  nu- 
merical systems  which  have  been  proposed  in  the  past  where  the 
supposed  relations  were  due  to  the  counting  of  blanks  which  do  not 
correspond  to  atomic  numbers.  This  peculiar  relationship  is  un- 
doubtedly connected  with  the  variations  in  structure  of  these  com- 
plex elements,  but  their  meaning  will  not  be  apparent  until  we 
know  more  in  regard  to  atomic  structure. 

*  If  elements  of  atomic  weights  two  and  three  are  ever  discovered  then  the 
zero  cycle  would  contain  2s  elements,  and  period  number  one  should  then  be 
said  to  begin  with  lithium.  Such  extrapolation,  however,  is  an  uncertain 
basis  for  the  prediction  of  such  elements. 


70  THEORETICAL  CHEMISTRY 

"The  first  cycle  of  two  short  periods  is  made  up  wholly  of  outer 
loop  or  main  group  elements.  Each  of  the  long  periods  of  the 
second  cycle  is  made  up  of  main  and  of  sub-group  elements,  and 
each  period  contains  one  eighth  group.  The  only  complete  very 
long  period  is  made  up  of  main  and  of  sub-group  elements,  con- 
tains one  eighth  group,  and  would  be  of  the  same  length  (18  ele- 
ments) as  the  long  periods  if  it  were  not  lengthened  to  32  elements 
by  the  inclusion  of  the  rare  earths. 

"The  first  long  period  is  introduced  into  the  system  by  the  in- 
sertion of  iron,  cobalt,  and  nickel,  in  its  center,  and  these  are  three 
elements  whose  atomic  numbers  increase  by  steps  of  one  while 
their  valence  remains  constant.  The  first  very  long  period  is 
formed  in  a  similar  way  by  the  insertion  of  the  rare  earths,  another 
set  of  elements  whose  atomic  numbers  increase  by  one  while  the 
valence  remains  constant. 

"In  this  periodic  table  the  maximum  valence  for  a  group  of 
elements  may  be  found  by  beginning  with  zero  for  the  zero  group 
and  counting  toward  the  front  for  positive  valence,  and  toward 
the  back  for  negative  valence. 

"  The  negative  valence  runs  along  the  spirals  toward  the  back 
as  follows :  — 

0  -1  -2  -3  -4 

Ne  F  O  N  C 

A  Cl  S  P  Si 

"Beginning  with  helium  the  relations  of  the  maximum  theoreti- 
cal valences  run  as  follows :  — 

Case  1.    He-F ....  0, 1,  2,  3,  4,  5,  6,  7,  but  does  not  rise  to  8.     Drops  by  7  to  0. 

Ne-Cl ...   0, 1,  2,  3,  4,  5,  6,  7,  but  does  not  rise  to  8.     Drops  by  7  to  0. 
Case  2.    A-Mn.  .  .   0,  1,  2,  3,  4,  5,  6,  7,  8,  8.     Drops  by  7  to  1. 

Fe,  Co,  Ni 
Case  1.    Cu-Br 
Case  2.    Kr-Ru,  Rh,  Pd. 

"In  the  third  increase,  the  group  number  and  maximum  valence 
of  the  group  rise  to  8,  three  elements  are  formed,  and  the  drop  is 
again  by  7  to  1. 

"Thus  in  every  case  when  the  valence  drops  back  the  drop  in 
maximum  group  valence  is  7,  either  from  7  to  0,  or  from  8  to  1. 


ATOMIC  STRUCTURE  71 

This  is  another  illustration  of  the  fact  that  the  eighth  group  is  a 
sub-group  of  the  zero  group.  The  valence  of  the  zero  group  is 
zero.  According  to  Abegg  the  contra-valence,  seemingly  not 
active  in  this  case,  is  eight. 

"  In  Fig.  13  the  table  is  divided  into  five  divisions  by  four 
straight  lines  across  the  base.  These  divisions  contain  the  fol- 
lowing groups :  — 

Division 0  1  2  3  4 

Groups 0,8  1,7  2,6  3,5  4,4 

"The  two  groups  of  any  division  are  said  to  be  complementary. 
It  will  be  seen  that  the  sum  of  the  group  numbers  in  any  division 
is  equal  to  8,  as  is  also  the  sum  of  the  maximum  valences.  The 
algebraic  sum  of  the  characteristic  valences  of  two  complementary 
groups  is  always  zero.  In  any  division  in  which  the  group  numbers 
are  very  different,  the  chemical  properties  of  the  elements  of  the  com- 
plementary main  groups  are  very  different,  but  when  the  group 
numbers  become  the  same,  the  chemical  properties  become  very  much 
alike.  Thus  the  greatest  difference  in  group  numbers  occurs  in 
division  8,  where  the  difference  is  8,  and  in  the  two  groups  there  is 
an  extreme  difference  in  chemical  properties,  as  there  is  also  in 
division  1  between  Groups  1  and  7. 

"Whenever  the  two  main  groups  of  a  division  are  very  different  in 
properties,  each  of  the  sub-groups  is  quite  different  Jrom  its  related 
main  group.  Thus  copper  in  Group  IB  is  not  very  closely  related 
to  potassium  Group  I A  in  its  properties,  and  manganese  is  not 
very  similar  to  chlorine,  but  as  the  group  numbers  approach  each 
other  the  main  and  sub-groups  become  much  alike.  Thus  scandium 
is  quite  similar  to  gallium  in  its  properties,  and  titanium  and  ger- 
manium are  very  closely  allied  to  silicon. 

"One  important  relation  is  that  on  the  outer  cylinder  the  main 
groups  I  A,  II  A,  III  A,  become  less  positive  as  the  group  number 
increases,  while  on  the  inner  loop  the  positive  character  increases  from 
Group  IB  to  IIB,  and  at  the  bottom  of  the  table  the  increase  from 
IIB  to  IIIB  is  considerable.  Thus  thallium  is  much  more  posi- 
tive than  mercury.  It  has  already  been  noted  that  in  the  case  of 
the  rare  earths  also  the  usual  rule  is  inverted,  that  is  the  basic 
properties  decrease  as  the  atomic  weight  increases." 


CHAPTER  VI. 
GASES. 

The  Gas  Laws.  Matter  in  the  gaseous  state  possesses  the 
property  of  filling  completely  and  to  a  uniform  density  any  avail- 
able space.  Among  the  most  pronounced  characteristics  of 
gases  are  lack  of  definite  shape  or  volume,  low  density  and  small 
viscosity.  The  laws  expressing  the  behavior  of  gases  under  differ- 
ent conditions  are  relatively  simple  and  to  a  large  extent  are 
independent  of  the  nature  of  the  gas.  The  temperature  and 
pressure  coefficients  of  all  gases  are  very  nearly  the  same. 

In  1662,  Robert  Boyle  discovered  the  familiar  law  that  at 
constant  temperature,  the  volume  of  a  gas  is  inversely  proportional 
to  the  pressure  upon  it.  This  may  be  expressed  mathematically 
as  follows :  — 

v  oc  -  (temperature  constant) 

where  v  is  the  volume  and  p  the  pressure. 

In  1801,  Gay-Lussac  discovered  the  law  of  the  variation  of  the 
volume  of  a  gas  with  temperature. 

This  law  may  be  formulated  thus:  —  At  constant  pressure,  the 
volume  of  a  gas  is  directly  proportional  to  its  absolute  temperature, 
or 

v  oc  T  (pressure  constant). 

There  are  three  conditions  which  may  be  varied,  viz.,  volume, 
temperature  and  pressure.  The  preceding  laws  have  dealt  with 
the  relation  between  two  pairs  of  the  variables  when  the  third 
is  held  constant.  There  remains  to  consider  the  relation  between 
the  third  pair  of  variables,  pressure  and  temperature,  the  volume 
being  kept  constant.  Evidently  a  necessary  corollary  of  the  first 
two  laws  is  that  at  constant  volume,  the- pressure  of  a  gas  is  directly 
proportional  to  its  absolute  temperature,  or 

p  oc  T7  (volume  constant). 
72 


GASES  73 

These  three  laws  may  be  combined  into  a  single  mathematical 
expression  as  follows  :  — 

v  oc  -  (T  const.)  law  of  Boyle, 

v  cc  T  (p  const.)  law  of  Gay-Lussac; 
combining  these  two  variations  we  have, 

T 

v  <x  —  , 

P 
or  introducing  a  proportionality  factor  k, 

i  T 

v  =  k  —  , 

P 
or 

vp  =  kT.  (1) 

If  the  temperature  of  the  gas  be  0°  (273°  absolute),  and  the  corre- 
sponding volume  and  pressure  v0  and  po  respectively,  then  (1) 
becomes 

vopQ  =  273  k, 

and 

k  - 
" 


273 

eliminating  the  constant  k  between  (1)  and  (2),  we  have 


For  any  one  gas  the  term  ^      is  a  constant.     If  v0  is  the  volume 


of  1  gram  of  gas  at  0°  and  76  cm.,  we  write 

vp  =  rT,  (3) 

where  v  is  the  volume  of  1  gram  of  gas  at  the  temperature  T  and 
the  pressure  p,  and  r  is  a  constant  called  the  specific  gas  constant. 
On  the  other  hand  when  VQ  denotes  the  volume  of  one  mol.  of  gas 

rO°  and  76  cm.  (22.4  liters),  the  equation  becomes 
vp  =  RT,  (4) 

aere  R  is  termed  the  molecular  gas  constant,  which  has  the  same 


74  THEORETICAL  CHEMISTRY 

value  for  all  gases.     If  M  is  the  molecular  weight  of  the  gas,  Mr 
=  R.     Equation  (4)  is  the  fundamental  gas  equation. 

Evaluation  of  the  Molecular  Gas  Constant.  Since  the  product 
of  p  and  v  represents  work,  and  T  is  a  pure  number,  R  must  be 
expressed  in  energy  units.  There  are  four  different  units  in  which 
the  molecular  gas  constant  is  commonly  expressed,  viz.,  (1)  gram- 
centimeters,  (2)  ergs,  (3)  calories,  and  (4)  liter-atmospheres. 

1.  R  in  gram-centimeters.     The  volume,  v,  of  1  mol.  of  gas  at 
0°  and  76  cm.  is  22.4  liters  or  22,400  cc.     The  pressure,  p,  is  76  cm. 
multiplied  by  13.59,  (the  density  of  mercury),  or  1033.3  grams  per 
square  centimeter.     Substituting  we  obtain 

fl=^o  =  1033.3^22,400  =  8 

2.  R  in  ergs.     To  convert  gram-centimeters  into  ergs  we  must 
multiply  by  the  acceleration  due  to  gravity,  g   --=  980.6  cm.  per 
sec.  per  sec.,  or 

R  =  84,760  X  980.6  =  83,150,000  ergs. 

3.  R  in  calories.     To  express  work  in  terms  of  heat,  we  must 
divide  by  the  mechanical  equivalent  of  heat,  or  since  1  calorie  is 
equivalent  to  42,640  gr.  cm.  or  41,830,000  ergs,  we  have 

R  =      ''        =  1.99  cal,  (approximately  2  cal.) 


4.  R  in  liter-atmospheres.  A  liter-atmosphere  may  be  defined 
as  the  work  done  by  1  atmosphere  on  a  square  decimeter  through 
a  decimeter.  If  p0  is  the  pressure  in  atmospheres,  and  VQ  is  the 
volume  in  liters,  we  have 


R  =          =      :    =0.0821  liter-atmosphere. 
J.          £l& 


Deviations  from  the  Gas  Laws.  Careful  experiments  by 
Amagat  *  and  others  on  the  behavior  of  gases  over  extended  ranges 
of  temperature  and  pressure  have  shown  that  the  fundamental 
gas  equation,  pv  =  RT,  is  not  strictly  applicable  to  any  one  gas, 
the  deviations  depending  upon  the  nature  of  the  gas  and  the 
conditions  under  which  it  is  observed.  It  has  been  shown  that 
the  gas  laws  are  more  nearly  obeyed  the  lower  the  pressure,  the 
*  Ann.  Chim.  phys.  (5)  19,  345  (1880). 


GASES 


75 


higher  the  temperature  and  the  further  the  gas  is  removed  from 
the  critical  state.  A  gas  which  would  conform  to  the  require- 
ments of  the  fundamental  gas  equation  is  called  an  ideal  or  per- 
fect gas.  Almost  all  gases  are  far  from  ideal  in  their  behavior. 
At  constant  temperature  the  product,  pv,  in  the  gas  equation  is 
constant,  so  that  if  we  plot  pressures  as  abscissae  and  the  corre- 


,20 


Ideal  gas 


cPj 


100 


200 


300 


P  (atmospheres) 
Fig.  14. 


spending  values  of  pv  as  ordinates,  for  an  ideal  gas  we  should 
obtain  a  straight  line  parallel  to  the  axis  of  abscissae,  as  shown  in 
Fig.  14.  The  results  obtained  by  Amagat  with  three  typical 
gases  are  also  shown  in  the  same  diagram.  It  will  be  apparent 
that  all  of  these  gases  depart  widely  from  ideal  behavior.  In 
the  case  of  hydrogen  pv  increases  continuously  with  the  pressure, 


76  THEORETICAL  CHEMISTRY 

while  with  nitrogen  and  carbon  dioxide  it  first  decreases,  attains 
to  a  minimum  value  and  beyond  that  point  increases  with  increas- 
ing pressure.  With  the  exception  of  hydrogen,  all  gases  show  a 
minimum  hi  the  curve,  thus  indicating  that  at  first  the  compress- 
ibility is  greater  than  corresponds  with  the  law  of  Boyle,  but 
diminishes  continuously  until,  for  a  short  range  of  pressure,  the  law 
is  followed  strictly:  beyond  this  point  the  compressibility  is  less 
than  Boyle's  law  requires. 

Hydrogen  is  exceptional  in  that  it  is  always  less  compressible 
than  the  law  demands.  This  is  true  for  all  ordinary  tempera- 
tures, but  it  is  highly  probable  that  at  extremely  low  temperatures 
the  curve  would  show  a  minimum.  The  two  curves  for  carbon 
dioxide  at  31°.5  and  100°  illustrate  the  fact  that  the  deviations 
from  the  gas  laws  become  less  as  the  temperature  increases.  The 
deviations  of  gases  from  the  laws  of  Boyle  and  Gay-Lussac,  as  well 
as  their  behavior  in  general,  may  be  satisfactorily  accounted  for 
on  the  basis  of  the  kinetic  theory. 

Kinetic  Theory  of  Gases.  The  first  attempt  to  explain  the 
properties  of  gases  on  a  purely  mechanical  basis  was  made  by 
Bernoulli  in  1738.  Subsequently,  through  the  labors  of  Kroenig, 
Clausius,  Maxwell,  Boltzmann  and  others,  his  ideas  were  developed 
into  what  is  known  today  as  the  kinetic  theory  of  gases.  Accord- 
ing to  this  theory,  gases  are  considered  to  be  made  up  of  minute, 
perfectly  elastic  particles  which  are  ceaselessly  moving  about 
with  high  velocities,  colliding  with  each  other  and  with  the  walls 
of  the  containing  vessel.  These  particles  are  identical  with  the 
molecules  defined  by  Avogadro.  The  volume  actually  occupied 
by  the  gas  molecules  is  supposed  to  be  much  smaller  than  the 
volume  filled  by  them  under  ordinary  conditions,  thus  allowing 
the  molecules  to  move  about  free  from  one  another's  influence 
except  when  they  collide.  The  distance  through  which  a  molecule 
moves  before  colliding  with  another  molecule  is  known  as  its  mean 
free  path.  In  terms  of  this  theory,  the  pressure  exerted  by  a  gas 
is  due  to  the  combined  effect  of  the  impacts  of  the  moving  molecules 
upon  the  walls  of  the  containing  vessel,  the  magnitude  of  the 
pressure  being  dependent  upon  the  kinetic  energy  of  the  mole- 
cules and  their  number. 


GASES 


77 


Derivation  of  the  Kinetic  Equation.  Starting  with  the  assump- 
tions already  made,  it  is  possible  to  derive  a  formula  by  means  of 
which  the  gas  laws  may  be  deduced.  Imagine  n  molecules,  each 
having  a  mass,  m,  confined  within  the  cubical  vessel  shown  in 
Fig.  15,  the  edge  of  which  has  a  length,  I.  While  the  different 
molecules  are  doubtless  moving  with  different  velocities,  there 
must  be  an  average  velocity  for  all  of  them.  Let  c  denote  this 
mean  velocity  of  translation.  The  molecules  will  impinge  upon 
the  walls  in  all  directions  but  the  velocity  of  each  may  be  resolved 
according  to  the  well-known  dynamical  principle  into  three  corn- 


Fig.  15. 

ponents,  x,  y  and  z,  parallel  to  the  three  rectangular  axes,  X,  Y 
and  Z.  The  analytical  expression  for  the  velocity  of  a  single 
molecule,  M,  is 


In  words,  this  means  that  the  effect  of  the  collision  of  the  molecule 
upon  the  wall  of  the  containing  vessel,  is  equivalent  to  the  com- 
bined effect  of  -successive  collisions  of  the  molecule  perpendicular 
to  the  three  walls  of  the  cubical  vessel  with  the  velocities  x,  y  and 
z  respectively.  Fixing  our  attention  upon  the  horizontal  com- 
ponent, the  molecule  will  collide  with  the  wall  with  a  velocity  x, 
and  owing  to  its  perfect  elasticity  it  will  rebound  with  a  velocity 


78  THEORETICAL  CHEMISTRY 

—  x,  having  suffered  no  loss  in  kinetic  energy.  The  momentum 
before  collision  was  mx  and  after  collision  it  will  be  —  mx,  the 
total  change  in  momentum  being  2  mx.  The  distance  between 
the  two  walls  being  I,  the  number  of  collisions  on  a  wall  in  unit 
time  will  be,  x/l,  and  the  total  effect  of  a  single  molecule  in  one 
direction  in  unit  time  will  be  2  mx  •  x/l  =  2  mx2/l.  The  same 
reasoning  is  applicable  to  the  other  components,  so  that  the  com- 
bined action  of  a  single  molecule  on  the  six  sides  of  the  vessel 
will  be 

2  m  f  „  ON      2  me2 

—  (z2  +  2/2  +  z2)  =  —  j  — 


O 

There  being  n  molecules,  the  total  effect  will  be  —  j  --     The 

entire  inner  surface  of  the  cubical  vessel  being  6  Z2,  the  pressure  p, 
on  unit  area,  will  be 

2  mnc2      1   mm2  . 


but  since  Z3  is  the  volume  of  the  cube,  which  we  will  denote  by  v, 
we  have 

1   mnc2 


or 

pv  =  -  •  mnc2  . 
o. 

This  is  the  fundamental  equation  of  the  kinetic  theory  of  gases. 
While  the  equation  has  been  derived  for  a  cubical  vessel,  it  is 
equally  applicable  to  a  vessel  of  any  shape  whatever,  since  the 
total  volume  may  be  considered  to  be  made  up  of  a  large  number 
of  infmitesimally  small  cubes,  for  each  of  which  the  equation  holds. 
Deductions  from  the  Kinetic  Equation.  Law  of  Boyle.  In 
the  fundamental  kinetic  equation,  pv  =  J  mnc2,  the  right-hand 
side  is  composed  of  factors  which  are  constant  at  constant  temper- 
ature, and  therefore  the  product,  pv,  must  be  constant  also  under 
similar  conditions.  This  is  clearly  Boyle's  law. 


GASES  79 

Law  of  Gay-Lussac.     The  kinetic  equation  may  be  written  in 
the  form 

2   1 

pv  =     ' 


The  kinetic  energy  of  a  single  molecule  being  represented  by 
1/2  me2,  the  total  kinetic  energy  of  the  molecules  of  the  gas  will 
be  1/2  mnc2.  Therefore  the  product  of  the  pressure  and  volume  of 
the  gas  is  equivalent  to  two-thirds  of  the  kinetic  energy  of  its  molecules., 
A  corollary  to  this  proposition  is  that  at  constant  pressure,  the 
average  kinetic  energy  of  the  molecules  in  equal  volumes  of  different 
gases  is  the  same.  The  law  of  Gay-Lussac  teaches  that  at  constant 
volume,  the  pressure  of  a  gas  is  directly  proportional  to  its  abso- 
lute temperature.  Taking  this  together  with  the  fact  that  the 
pressure  of  a  gas  at  constant  volume  is  directly  proportional  to 
the  mean  kinetic  energy  of  its  molecules,  it  follows  that  the  mean 
kinetic  energy  of  the  molecules  of  a  gas  is  directly  proportional  to  its 
absolute  temperature.  Thus  we  see  that  the  absolute  temperature  of 
a  gas  is  a  measure  of  the  mean  kinetic  energy  of  its  molecules.  This 
deduction  is  partially  based  upon  the  experimentally-determined 
law  of  Gay-Lussac.  Having  obtained  a  definiton  of  temperature 
in  terms  of  kinetic  energy,  it  is  easy  to  derive  Gay-Lussac's  law 
from  the  fundamental  kinetic  equation.  Writing  the  equation  in 
the  form 

2   1 


it  is  apparent  that  pv  is  directly  proportional  to  the  total  kinetic 
energy  of  the  gas  molecules,  or  in  other  words,  is  directly  propor- 
tional to  its  absolute  temperature,  which  is  the  most  general 
statement  of  Gay-Lussac's  law. 

Hypothesis  of  Avogadro.     If  equal  volumes  of  two  different  gases 
measured  under  the  same  pressure,  we  will  have 


pv  =  l/3nimiCi2  =  l/S^ra^2,  (1) 


f 
ere  n\  and  nz,  mi  and  m2,  and  Ci  and  c2  denote  the  number,  mass 
1  velocity  of  the  molecules  in  the  two  gases.     If  the  gases  are 


80  THEORETICAL  CHEMISTRY 

measured  at  the  same  temperature,  the  molecules  of  each  possess 
the  same  mean  kinetic  energy,  or 

l/2mici2  =  l/2m2c22.  (2) 

Dividing  equation  (1)  by  equation  (2),  we  have 

HI  =  nz, 

or  under  the  same  conditions  of  temperature  and  pressure  equal 
volumes  of  the  two  gases  contain  the  same  number  of  molecules. 
.This  is  the  hypothesis  of  Avogadro. 

Law  of  Graham.     If  the  fundamental  kinetic  equation  be  solved 
for  c,  we  have 


mn 


but  v/mn  =  l/d,  where  d  is  the  density  of,  the  gas,  and  therefore 
we  may  write 


If  the  pressure  remains  constant  it  is  evident  that  the  mean  veloc- 
ities of  the  molecules  of  two  gases  are  inversely  proportional  to 
the  square  roots  of  their  densities,  a  law  which  was  first  enunciated 
by  Graham  in  1833  as  the  result  of  his  experiments  on  gaseous 
diffusion. 

Mean  Velocity  of  Translation  of  a  Gaseous  Molecule.  By 
substituting  appropriate  values  for  the  various  magnitudes  in  the 
equation 


c  — 

*   mn 

it  is  possible  to  calculate  the  mean  velocity  of  the  molecules  of  any 
gas.  Thus,  for  the  gram-molecule  of  hydrogen  at  0°  and  76  cm. 
pressure,  p  =  76  X  13.59  =  1033.3  gr.  per  sq.  cm.  =  1033.3  X  980.6 
dynes  per  sq.  cm.,  v  =  22,400  cc.,  and  mn  =  2.016  gr. 

Substituting  these  values  in  the  above  equation  we  have, 

4  /3  X  1033.3  X  980.6  X  22,400 
c  ==  y  -  2p1ft   -  -  =  183,780  cm.  per  sec. 


GASES  81 

Thus  at  0°  the  molecule  of  hydrogen  moves  with  a  speed  slightly 
greater  than  one  mile  per  second.  This  enormous  speed  is  only 
attained  along  the  mean  free  path,  the  frequent  collisions  with 
other  molecules  rendering  the  actual  speed  much  less  than  that 
calculated. 

Equation  of  van  der  Waals.  As  has  been  pointed  out  in  a 
previous  paragraph,  the  gas  laws  are  merely  limiting  laws  and 
while  they  hold  quite  well  up  to  pressures  of  about  2  atmospheres, 
above  this  pressure  the  differences  between  the  observed  and  cal- 
culated values  become  steadily  larger.  In  the  case  of  hydrogen, 
Natterer  was  the  first  to  show  that  the  product  of  pressure  and 
volume  is  invariably  higher  than  it  should  be.  A  possible  explana- 
tion of  this  departure  from  the  gas  laws  was  offered  by  Budde, 
who  proposed  that  the  volume,  v,  in  the  equation  pv  =  R T,  should 
be  corrected  for  the  volume  occupied  by  the  molecules  them- 
selves. If  this  volume  correction  be  denoted  by  6,  then  the  gas 
equation  becomes 

p  (v  -  6)  =  RT, 

where  6  is  a  constant  for  each  gas.  Budde  calculated  the  value 
of  b  for  hydrogen  and  found  it  to  remain  constant  for  pressures 
varying  from  1000  to  2800  meters  of  mercury. 

While  Budde's  modification  of  the  gas  equation  is  quite  satis- 
factory in  the  case  of  hydrogen,  it  fails  when  applied  to  other  gases. 
In  general,  the  compressibility  at  low  pressures  is  considerably 
greater  than  can  be  accounted  for  by  Boyle's  law.  The  compressi- 
bility reaches  a  minimum  value,  and  then  increases  rapidly  so  that 
pv  passes  through  the  value  required  by  the  law.  This  suggests 
that  there  is  some  other  correction  to  be  applied  in  addition  to 
the  volume  correction  introduced  into  the  gas  equation  by  Budde. 
van  der  Waals  pointed  out  in  1879,  that  in  the  deduction  of  Boyle's 
law  by  means  of  the  fundamental  kinetic  equation,  the  tacit 
assumption  is  made  that  the  molecules  exert  no  mutual  attraction. 
While  this  assumption  is  undoubtedly  justifiable  when  the  gas  is 
subjected  to  a  very  low  pressure,  it  no  longer  remains  so  when 
the  gas  is  strongly  compressed.  A  little  consideration  will  make 
it  apparent  that  when  increased  pressure  is  applied  to  a  gas,  the 


82  THEORETICAL  CHEMISTRY 

resulting  volume  will  become  less  than  that  calculated,  owing  to 
molecular  attraction.  In  other  words  the  molecular  attraction 
and  the  applied  pressure  act  in  the  same  direction  and  the  gas 
behaves  as  if  it  were  subjected  to  a  pressure  greater  than  that 
actually  applied.  Van  der  Waals  showed  that  this  correction  is 
inversely  proportional  to  the  square  of  the  volume,  and  since  it 
augments  the  applied  pressure  the  expres^on  p  -f-  a/vz  is  sub- 
stituted for  p  in  the  gas  equation,  a  being  the  constant  of  molecular 
attraction.  The  corrected  equation  then  becomes 

(p  +  aA2)  (v  -  6)  =  RT. 

This  is  known  as  the  equation  of  van  der  Waals.  It  is  applicable 
not  only  to  strongly  compressed  gases,  but  also  to  liquids  as  well. 
While  it  will  be  given  detailed  consideration  in  a  subsequent  chapter, 
it  may  be  of  interest  to  point  out  at  this  time  the  satisfactory  ex- 
planation which  it  offers  of  the  experimental  results  of  Amagat, 
to  which  we  have  already  made  reference,  (page  72).  When  v  is 
large,  both  b  and  a/v2  become  negligible,  and  van  der  Waals' 
equation  reduces  to  the  simple  gas  equation,  pv  —  RT.  We  may 
predict,  therefore,  that  any  influence  tending  to  increase  v  will 
cause  the  gas  to  approach  more  nearly  to  the  ideal  condition.  This 
is  in  accord  with  the  results  of  Amagat's  experiments,  which  show 
that  an  increase  of  temperature  at  constant  pressure,  or  a  diminu- 
tion of  pressure  at  constant  temperature,  causes  the  gas  to  tend  to 
follow  the  simple  gas  laws.  The  equation  also  offers  a  satisfactory 
explanation  of  the  exceptional  behavior  of  hydrogen  when  it  is 
subjected  to  pressure.  As  we  have  seen,  pv  for  all  gases,  except 
hydrogen,  diminishes  at  first  with  increasing  pressure,  reaches  a 
minimum  value,  and  then  increases  regularly.  Since  the  volume 
correction  in  van  der  Waals'  equation  acts  in  opposition  to  the 
attraction  correction,  it  is  apparent  that  at  low  pressures  the  effect 
of  attraction  preponderates,  while  at  high  pressures  the  volume 
correction  is  relatively  of  more  importance.  At  some  intermediate 
pressure  the  two  corrections  counterbalance  each  other,  and  it  is 
at  this  point  that  the  gas  follows  Boyle's  law  strictly.  The 
exceptional  behavior  of  hydrogen  may  be  accounted  for  by  making 
the  very  plausible  assumption  that  the  attraction  correction  is 


GASES  83 

negligible  at  all  pressures  in  comparison  with  the  volume  correc- 
tion. 

Vapor  Density  and  Molecular  Weight.  As  has  been  pointed  out 
in  an  earlier  chapter,  when  a  substance  can  be  obtained  in  the  gas- 
eous state,  the  determination  of  its  molecular  weight  resolves  itself 
into  finding  the  mass  of  that  volume  of  vapor  which  will  occupy 
22.4  liters  at  0°  and  76  cm.  It  is  inconvenient  to  weigh  a  volume 
of  gas  or  vapor  under  standard  conditions  of  temperature  and 
pressure,  but  by  means  of  the  gas  laws  the  determination  made 
at  any  temperature  and  under  any  pressure  can  be  reduced  to 
standard  conditions.  For  example,  suppose  v  cc.  of  gas  are  found 
to  weigh  w  grams  at  t°  and  p  cm.  pressure,  then  the  weight  in  grams 
of  22.4  liters  or  22,400  cc.  at  0°  and  76  cm.  will  be  given  by  the  fol- 
lowing proportion,  in  which  M  denotes  the  molecular  weight  of  the 
substance  :  — 

pv  ,,    76  X  22,400 

--- 


or 

w  X  76  X  22,400  X  (t  +  273) 


M  = 


273  pv 


The  determination  of  vapor  density  may  be  effected  in  either  of 
two  ways;  (1)  we  may  determine  the  mass  of  a  known  volume  of 
vapor  under  definite  conditions  of  temperature  and  pressure,  or 
(2)  we  may  determine  the  volume  of  a  known  mass  under  definite 
conditions  of  temperature  and  pressure.  There  are  a  variety  of 
methods  for  the  determination  of  vapor  density;  but  for  our  pur- 
pose it  will  be  necessary  to  describe  but  two  of  them.  In  the 
method  of  Regnault  the  mass  of  a  definite  volume  of  vapor  is 
determined,  while  in  the  method  due  to  Victor  Meyer  we  measure 
the  volume  of  a  known  mass. 

Method  of  Regnault.  In  this  method  which  is  especially  adapted 
to  permanent  gases,  use  is  made  of  two  spherical  glass  bulbs 
(Fig.  16)  of  approximately  the  same  capacity,  each  bulb  being 
provided  with  a  well-ground  stop-cock.  By  means  of  an  airpump 
one  bulb  in  evacuated  as  completely  as  possible,  and  is  then  filled, 
at  definite  temperature  and  pressure,  with  the  gas  whose  density 


84  THEORETICAL  CHEMISTRY 

is  to  be  determined.  The  stop-cock  is  then  closed  and  the  bulb 
weighed,  the  second  bulb  being  used  as  a  counterpoise.  The  use 
of  the  second  bulb  is  largely  to  avoid  the 
troublesome  corrections  for  air  displacement 
and  for  moisture,  each  bulb  being  affected  in 
the  same  way  and  to  nearly  the  same  extent. 
The  volume  of  the  bulb  may  be  obtained  by 
weighing  it  first  evacuated,  and  then  filled  with 
distilled  water  at  known  temperature.  From 
these  results  we  may  calculate  the  mass  per  unit 
of  volume;  or  we  may  substitute  the  values  of 
w,  v}  p  and  t  in  the  above  formula  and  calcu- 
late M,  the  molecular  weight.  This  method 
was  used  by  Morley  *  in  his  epoch-making  re- 
search on  the  densities  of  hydrogen  and  oxygen. 
Method  of  Victor  Meyer.  In  the  method 
of  Victor  Meyer,  a  weighed  amount  of  the 
substance  is  vaporized,  and  the  volume  which  it  would  have 
occupied  at  the  temperature  of  the  room  and  under  existing 
barometric  pressure  is  determined.  The  apparatus  of  Meyer, 
shown  in  Fig.  17,  consists  of  an  inner  glass  tube  A,  about  1 
cm.  in  diameter  and  75  cm.  in  length.  This  tube  is  expanded 
into  a  bulb  at  the  lower  end,  while  at  the  top  it  is  slightly  en- 
larged and  is  furnished  with  two  side  tubes  C  and  E.  The  tube 
A  is  suspended  inside  a  heating  jacket  B,  containing  some  liquid 
the  boiling  point  of  which  is  about  20°  higher  than  the  vaporizing 
temperature  of  the  substance  whose  vapor  density  is  to  be  de- 
termined. The  side  tube  E  dips  beneath  the  surface  of  water  in  a 
pneumatic  trough  G,  and  serves  to  convey  the  air  displaced  from 
A  to  the  eudiometer  F.  By  means  of  the  side  tube  C,  and  the  glass- 
rod  D,  the  small  bulb  containing  the  substance  can  be  dropped  to 
the  bottom  of  A.  To  carry  out  a  determination  of  vapor  density 
with  this  apparatus,  the  liquid  in  B  is  heated  to  boiling  and 
the  sealed  bulb  V,  containing  a  weighed  amount  of  the  substance, 
is  placed  in  position  on  the  rod  D,  the  corks  being  tightly  inserted. 

*  Smithsonian  Contributions  to  Knowledge,  (1895). 


GASES 


85 


When  bubbles  of  air  cease  to  issue  from  E  in  the  pneumatic  trough, 
showing  that  the  temperature  within  A  is  constant,  the  eudiometer 
F,  full  of  water,  is  placed  over  the  mouth  of  E,  and  the  bulb  V  is 
allowed  to  drop  by  drawing  aside  the  rod  D.  Air  bubbles  immedi- 
ately begin  to  issue  from  E  and  to  collect  in  the  eudiometer.  When 
the  air  ceases  to  collect,  the  eudiometer  is  closed  by  the  thumb  and 


Fig.  17. 

is  removed  to  a  large  cylinder  of  water  where  it  is  allowed  to  stand 
long  enough  to  acquire  the  temperature  of  the  room.  It  is  then 
raised  or  lowered  until  the  level  of  water  inside  and  outside  is 
the  same,  when  the  volume  of  air  is  carefully  read  off.  In  this 
method,  the  substance  on  vaporizing  displaces  an  equal  volume 
of  air  which  is  collected  and  measured,  this  observed  volume  being 


86  THEORETICAL  CHEMISTRY 

that  which  the  vapor  would  occupy  after  reduction  to  the  condi- 
tions under  which  the  air  is  measured.  It  is  evident  that  in  this 
method  we  do  not  require  a  knowledge  of  the  temperature  at 
which  the  substance  vaporizes.  Since  the  air  is  measured  over 
water,  the  pressure  to  which  it  is  subjected  is  that  of  the  atmos- 
phere diminished  by  the  vapor  pressure  of  water  at  the  temperature 
of  the  experiment.  The  method  of  calculating  molecular  weights 
from  the  observations  recorded  may  be  illustrated  by  the  follow- 
ing example:  —  0.1  gram  of  benzene  (C6H6)  was  weighed  out,  and 
when  vaporized,  32  cc.  of  air  were  collected  over  water  at  17°  and 
750  mm.  pressure.  The  vapor  pressure  of  water  at  17°  is  14.4 
mm.,  and  the  actual  pressure  exerted  by  the  gas  is  750  —  14.4  = 
735.6  mm.  Substituting  in  the  proportion 

pv  .    760  X  22,400 

~- 


and  solving  for  M  we  have 

0.1  X  760  X  22,400  X  (17  +  273) 
273  X  735.6  X  32 

The  result  agrees  fairly  well  with  the  molecular  weight  of  benzene 
(78.05)  calculated  from  the  formula. 

Unless  a  vapor  follows  the  gas  laws  very  closely,  the  value  of  the 
molecular  weight  obtained  by  the  method  of  Victor  Meyer  will  be 
only  approximate,  but  this  approximate  value  will  be  sufficiently 
near  to  the  true  molecular  weight  to  enable  us  to  choose  between 
the  simple  formula  weight,  given  by  chemical  analysis,  and  some 
multiple  of  it. 

Results  of  Vapor-Density  Determinations.  As  the  result  of 
numerous  vapor-density  determinations  extending  over  a  wide 
range  of  temperatures,  much  important  data  has  been  collected 
concerning  the  number  of  atoms  contained  in  the  molecules  of  a 
large  number  of  chemical  compounds.  The  molecular  weights 
of  most  of  the  elementary  gases  are  double  their  atomic  weights, 
showing  that  their  molecules  are  diatomic.  In  like  manner  the 
molecular  weights  of  mercury,  zinc,  cadmium  and,  in  fact,  all  of 
the  vaporizable  metallic  elements  have  been  found  to  be  identi- 
cal with  their  atomic  weights.  The  molecules  of  sulphur, 


GASES  87 

arsenic,  phosphorus  and  iodine  are  polyatomic,  if  they  are  not 
heated  to  too  high  a  temperature.  The  investigations  of  Meyer 
and  others  have  shown  that  the  vapor  densities  of  a  large  number 
of  substances  diminish  as  the  temperature  is  increased.  In  other 
words  as  the  temperature  is  raised  the  number  of  atoms  contained 
in  the  molecules  decreases.  The  molecular  weight  of  sulphur,  cal- 
culated from  its  vapor  density  at  temperatures  below  500°,  corre- 
sponds to  the  formula  S8.  If  the  vapor  of  sulphur  is  heated  to 
1100°,  the  molecular  weight  corresponds  to  the  formula  S2.  In 
fact,  sulphur  in  the  form  of  vapor  may  be  represented  by  the  formu- 
las S8)  84,  $2,  or  even  S  according  to  the  temperature  at  which  its 
vapor  density  is  determined.  Iodine  behaves  similarly,  the  mole- 
cules being  diatomic  between  200°  and  600°,  while  at  temperatures 
above  1400°  the  vapor  density  has  about  one-half  its  value  at  the 
lower  temperature,  showing  a  complete  breaking  down  of  the  dia- 
tomic molecules  into  single  atoms.  Heating  to  yet  higher  tem- 
peratures has  failed  to  reveal  any  further  decomposition.  This 
phenomenon  is  not  confined  to  the  molecules  of  the  elements  alone, 
but  is  also  met  with  in  the  case  of  the  molecules  of  chemical  com- 
pounds. The  vapor  density  of  arsenious  oxide  between  500°  and 
700°  corresponds  to  the  formula  As406.  As  the  temperature  is 
raised,  the  vapor  density  becomes  steadily  smaller  until,  at  1800°,  the 
calculated  molecular  weight  corresponds  to  the  formula  As203.  In 
like  manner  ferric  and  aluminium  chlorides  have  been  shown  to 
have  molecular  weights  at  low  temperatures  corresponding  to  the 
formulas,  Fe2Cl6  and  A12C16.  The  commonly-used  formulas,  FeCU 
and  A1C1.3,  represent  their  molecular  weights  at  high  temperatures 
only.  The  experimental  difficulties  attending  vapor  density  de- 
terminations increase  as  the  temperature  is  raised,  owing  chiefly  to 
the  deformation  of  the  apparatus  when  the  material  of  which  it  is 
constructed  approaches  its  melting  point.  Glass  which  can  be  used 
at  relatively  low  temperatures  only,  has  been  replaced  by  specially 
resistant  varieties  of  porcelain  which  may  be  used  up  to  tempera- 
tures of  1500°  or  1600°.  Platinum  vessels  retain  their  shape  up  to 
temperatures  between  1700°  and  1800°.  Measurements  up  to 
2000°  have  recently  been  effected  by  Nernst  and  his  pupils.*  In 

*  Wartenberg.  Zeit.  anorg.  Chem.,  56,  320  (1907). 


88  THEORETICAL  CHEMISTRY 

their  experiments  use  was  made  of  a  vessel  of  iridium,  the  inside 
and  outside  of  which  was  surrounded  with  a  cement  of  magnesia 
and  magnesium  chloride,  the  entire  apparatus  being  heated  electri- 
cally. With  this  apparatus  they  snowed  that  the  molecular 
weight  of  sulphur  between  1800°  and  2000°  is  48,  indicating  that 
the  diatomic  molecule  is  approximately  50  per  cent  broken  down 
into  single  atoms. 

Abnormal  Vapor  Densities.  In  all  of  the  cases  cited  above 
the  molecular  weight  calculated  from  the  vapor  density  corre- 
sponds either  with  the  simple  formula  weight,  as  determined  by 
chemical  analysis,  or  with  a  multiple  thereof.  In  no  case  is  there 
any  evidence  of  a  breaking  down  of  the  simple  molecule  into  its 
constituents.  Substances  are  known,  however,  the  molecular 
weights  of  which,  calculated  from  their  vapor  densities,  are  less 
than  the  sum  of  the  atomic  weights  of  their  constituents.  For 
example,  the  vapor  density  of  ammonium  chloride  was  found  to 
be  0.89,  while  that  corresponding  to  the  formula  NH4C1  should  be 
1.89.  Similar  results  have  been  obtained  with  phosphorus  penta- 
chloride,  nitrogen  peroxide,  chloral  hydrate  and  numerous  other 
substances.  The  phenomenon  can  be  explained  in  either  of  the  two 
following  ways:  (1)  that  the  molecule  has  undergone  a  complete 
disruption,  or  (2)  that  the  substance  does  not  follow  the  law  of 
Avogadro.  Until  the  former  explanation  was  shown  to  be  correct, 
the  latter  was  accepted  and  for  a  time  the  law  of  Avogadro  fell  into 
disrepute.  In  1857,  Deville  showed  that  numerous  chemical  com- 
pounds are  broken  down  or  "dissociated"  at  high  temperatures. 
Shortly  afterward  Kopp  suggested  that  the  abnormal  vapor 
densities  of  such  substances  as  ammonium  chloride,  phosphorus 
pentachloride,  etc.,  might  be  due  to  thermal  dissociation.  If 
ammonium  chloride  underwent  complete  dissociation,  one  molecule 
of  the  salt  would  yield  one  molecule  of  ammonia  and  one  molecule 
of  hydrochloric  acid  gas,  and  the  vapor  density  of  the  resulting 
mixture  would  be  one-half  of  that  of  the  undissociated  substance, 
a  deduction  in  complete  agreement  with  the  results  of  experiment. 
It  remained  to  prove  that  the  products  of  this  supposed  dissocia- 
tion were  actually  present. 

The  first  to  offer  an  experimental  demonstration  of  the  simul- 


GASES 


89 


taneous  formation  of  ammonia  and  hydrochloric  acid,  when  ammon- 
ium chloride  is  heated,  was  Pebal.*  The  apparatus  which  he  de- 
vised for  this  purpose  is  shown  in  Fig.  18.  It  consisted  of  two  tubes, 
T  and  t,  the  latter  being  placed  within  the  former  as  indicated  in 
the  sketch.  Near  the  top  of  the  inner  tube,  which  was  drawn  down 
to  a  smaller  diameter,  was  a  porous  plug  of  asbestos,  C,  upon  which 
was  placed  a  little  ammonium  chloride.  A  stream  of  .dry  hydro- 


Hydrogen 


Hydrogen 


Fig.  18. 


gen  was  passed  through  the  apparatus  by  means  of  the  tubes  A  and 
By  the  former  entering  the  outer  tube  and  the  latter  the  inner 
tube.  The  entire  apparatus  was  heated  to  a  temperature  above 
that  necessary  to  vaporize  the  ammonium  chloride.  If  the  salt 
undergoes  dissociation  into  ammonia  and  hydrochloric  acid,  the 
former  being  less  dense  than  the  latter,  would  diffuse  more 
rapidly  through  the  plug  C  and  the  vapor  below  the  plug  would 

*  Lieb.  Ann.,  123,  199  (1862). 


90 


THEORETICAL  CHEMISTRY 


be  relatively  richer  in  ammonia  than  the  vapor  above  it.  The 
current  of  hydrogen  through  B  would  therefore  sweep  out  from 
the  lower  part  of  t  an  excess  of  ammonia,  while  the  current  through 
A  would  carry  out  from  T  an  excess  of  hydrochloric  acid.  By 
holding  strips  of  moistened  litmus  paper  in  the  currents  of  gas 
issuing  from  E  and  F,  it  was  possible  for  Pebal  to  test  the  correct- 
ness of  Kopp's  idea.  He  found  that  the  gas  issuing  from  E  had 
an  acid  reaction  while  that  escaping  from  F  had  an  alkaline  reac- 
tion. It  would  at  first  sight  appear  that  Pebal  had  demonstrated 


Nitrogen. 


Fig.  19; 


beyond  question  that  ammonium  chloride  undergoes  dissociation 
into  ammonia  and  hydrochloric  acid. 

It  was  pointed  out,  however,  that  Pebal  had  heated  the  ammon- 
ium chloride  in  contact  with  a  foreign  substance,  asbestos,  and 
that  this  might  have  acted  as  a  catalyst,  promoting  the  decomposi- 
tion into  ammonia  and  hydrochloric  acid.  This  objection  was 
removed  by  the  ingenious  experiment  of  Than.*  He  devised  a 
modification  of  Pebal's  apparatus,  as  shown  in  Fig.  J9.  In  the 
horizontal  tube,  AB,  the  ammonium  chloride  was  placed  at  F  and  a 

*  Lieb.  Ann.,  131,  129  (1864). 


GASES  91 

porous  plug  of  compressed  ammonium  chloride  was  introduced  at 
G.  The  tube  was  heated  and  nitrogen  passed  in  at  C.  The 
reactions  of  the  currents  of  gas  issuing  at  D  and  E  were  tested 
with  litmus  as  in  PebaPs  experiment  and  it  was  found  that  the 
gas  escaping  from  D  was  alkaline,  while  that  issuing  from  E  was 
acid.  This  experiment  proved  beyond  question  that  the  vapor 
of  ammonium  chloride  is  thermally  dissociated  into  ammonia 
and  hydrochloric  acid.  Experiments  on  other  substances  whose 
vapor  densities  are  abnormally  small  show  that  a  similar  explan- 
ation is  applicable,  and  thus  furnish  a  confirmation  of  the  law  of 
Avogadro. 

Calculation  of  the  Degree  of  Dissociation.  Since  the  density 
of  a  dissociating  vapor  decreases  with  increase  in  temperature, 
it  is  important  to  be  able  to  calculate  the  degree  of  dissocation  at 
any  one  temperature.  This  is  clearly  equivalent  to  ascertaining 
the  extent  to  which  the  reaction 

NH4C1<=±NH3  +  HC1 

has  proceeded  from  left  to  right.  This  can  be  determined  easily 
from  the  relation  of  vapor  density  to  dissociation.  If  we  start 
with  one  molecule  of  gas  and  let  a  represent  the  percentage  dis- 
sociation, then  1  —  a  will  denote  the  percentage  remaining  un- 
dissociated.  If  one  molecule  of  gas  yields  n  molecules  of  gaseous 
products,  the  total  number  of  molecules  present  at  any  time  will 
be 

(1  -  a)  +  na  =  1  +  (n  -  1)  a. 

The  ratio  1  :  1  -f  (n  —  1)  a  will  be  the  same  as  the  ratio  of  the 
density  d2  of  the  dissociated  gas  to  its  density  in  the  undissociated 
state  di,  or 


Th 


solving  this  proportion  for  a,  we  have 


(n  -  1)  d2 

e  vapor  density  of  nitrogen  peroxide  has  been  measured  by  E. 
and  L.  Natanson,*  and  the  degree  of  dissociation  at  the  different 

*  Wied.  Ann.,  24,  454  (1885);  27,  606  (1886). 


92 


THEORETICAL  CHEMISTRY 


temperatures  calculated  by  means  of  the  preceding  formula.     The 
following  table  gives  their  results. 

The  course  of  the  dissociation  is  shown  in  the  accompanying 
illustration,  Fig.  20,  in  which  the  abscissae  represent  temperature 
and  the  ordinates,  percentage  dissociation.  It  will  be  observed 


10 


20 


60 


80  100 

Temperature 


Fig.  20. 


that  the  dissociation  of  nitrogen  peroxide  is  at  first  nearly  pro- 
portional to  the  temperature.  It  then  increases  more  rapidly 
until,  when  about  four-fifths  of  the  molecules  of  N204  are  broken 
down,  the  dissociation  proceeds  slowly  to  completion. 

Specific  Heat.  The  addition  of  heat  energy  to  a  body  causes 
its  temperature  to  rise.  The  ratio  of  the  amount  of  heat  supplied 
to  the  resulting  rise  in  temperature  is  called  the  heat  capacity  of 
the  body;  obviously  its  value  is  dependent  upon  the  initial  temper- 


GASES 


93 


DISSOCIATION  OF  NITROGEN  PEROXIDE,   N2O*. 

ATMOSPHERIC  PRESSURE. 
(Density  of  N/)4  =  3.18;    of  N02+NO2  =  1.59;  ofair  =  1.00) 


Temperature, 
(degrees) 

Density  of  Gas. 

Percentage  Dis- 
sociation. 

26.7 

2.65 

19.96 

35.4 

2.53 

25.65 

39.8 

2.46 

29.23 

49.6 

2.27 

40.04 

60.2 

2.08 

52.84 

70.0 

1.92 

65.57 

80.6 

1.80 

76.61 

90.0 

1.72 

84.83 

100.1 

1.68 

89.23 

111.3 

1.65 

92.67 

121.5 

1.62 

96.23 

135.0 

1.60 

98.69 

154.0 

1.58 

100.00 

ature  of  the  body.  The  specific  heat  of  a  substance  may  be  defined 
as  the  heat  capacity  of  unit  mass  of  the  substance.  If  dt  represents 
the  increment  of  temperature  due  to  the  addition  of  dQ  units  of 
heat  energy  to  m  grams  of  any  substance,  then  its  specific  heat,  c, 
will  be  given  by  the  equation 

c  —  _ •  — — • 
m    dt 

Specific  Heat  at  Constant  Pressure  and  Constant  Volume.     It 

is  well  known  that  the  specific  heat  of  a  gas  depends  upon  the 
conditions  under  which  it  is  determined.  If  a  definite  mass  of 
gas  is  heated  under  constant  pressure,  the  value  of  the  specific 
heat,  cp)  is  different  from  the  value  of  the  specific  heat,  cv,  ob- 
tained when  the  pressure  varies  and  the  volume  remains  con- 
stant. The  value  of  cp  is  invariably  greater  than  that  of  cv. 
When  heat  is  supplied  to  a  gas  at  constant  pressure  not  only  does 
its  temperature  rise,  but  it  also  expands,  and  thus  does  external 
work.  On  the  other  hand,  if  the  gas  be  heated  in  such  a  way  that 
its  volume  cannot  change,  none  of  the  heat  supplied  will  be  used 
in  doing  external  work,  and  consequently  its  heat  capacity  will 


94  THEORETICAL  CHEMISTRY 

be  less  than  when  it  is  heated  under  constant  pressure.  The 
recognition  by  Mayer  in  1841  of  the  cause  of  this  difference  between 
the  two  specific  heats  of  a  gas  led  him  to  his  celebrated  calculation 
of  the  mechanical  equivalent  of  heat,  and  the  enunciation  of  the 
first  law  of  thermodynamics.  Mayer  observed  that  the  differ- 
ence between  the  quantity  of  heat  necessary  to  raise  the  temper- 
ature of  1  gram  of  air  1°  C.  at  constant  pressure,  and  at  constant 
volume  respectively,  was  0.0692  calorie,  or 

Cp  -  Cv  =  0.0692  cal. 

That  is  to  say,  0.0692  calorie  is  the  amount  of  heat  energy  which 
is  equivalent  to  the  work  required  to  expand  1  gram  of  air  1/273 
of  its  volume  at  0°.  Imagine  1  gram  of  air  at  0°  enclosed  within 
a  cylinder  having  a  cross-section  of  one  square  centimeter,  and 
furnished  with  a  movable,  frictionless  piston.  Since  1  gram  of 
air  under  standard  conditions  of  temperature  and  pressure  occu- 
pies 773.3  cc.,  the  distance  between  the  piston  and  the  bottom  of 
the  cylinder  will  be  773.3  cm.  If  the  temperature  be  raised  from 
0°  to  1°,  the  piston  will  rise  1/273  X  773.3  =  2.83  cm.,  and  since 
the  pressure  of  the  atmosphere  is  1033.3  grams  per  square  centi- 
meter, the  external  work  done  by  the  expanding  gas  will  be 

1033.3  X  2.83  =  2924.3  gr.  cm. 

This  is  evidently  equivalent  to  0.0692  calorie  and  therefore,  the 
equivalent  of  1  calorie  in  mechanical  units,  J,  will  be 

= 42'258  gr- cm- 

a  value  agreeing  very  well  with  the  best  recent  determinations  of 
the  mechanical  equivalent  of  heat. 

The  difference  between  the  two  specific  heats  may  be  easily 
calculated  in  calories  from  the  fundamental  gas  equation.  Start- 
ing with  1  mol.  of  gas,  and  remembering  that  when  a  gas  expands 
at  constant  pressure,  the  product  of  pressure  and  change  in  volume 
is  a-  measure  of  the  work  done,  we  have,  at  temperature  TI°, 

pvi  =  RT1} 
where  Vi  is  the  molecular  volume.     Raising  the  temperature  to 


GASES  95 

TZ°J  the  corresponding  molecular  volume  being  vz,  we  have  for 
the  work  done  during  expansion 

p  (v*  -vl)=R  (T2  -  Ti). 
If  T2  —  TI  =  1°,  then  the  equation  reduces  to 

p  (vz  -  vi)  =  R. 

Since  the  difference  between  the  molecular  heats  *  at  constant 
pressure  and  constant  volume  is  equivalent  to  the  external  work 
involved  when  the  temperature  of  1  mol.  of  gas  is  raised  1°,  we  have 

M  (cp  —  cv)  =  p  (vz  —  vi)j 

where  M  is  the  molecular  weight  of  the  gas;  and  therefore 
M  (cp  —  cv)  =  R  =  2  calories. 

In  words,  the  difference  of  the  molecular  heats  of  any  gas  at 
constant  pressure  and  at  constant  volume  is  2  calories.  The 
specific  heat  of  a  gas  at  constant  pressure  can  be  readily  deter- 
mined, by  passing  a  definite  volume  of  the  gas,  heated  under  con- 
stant pressure  to  a  known  temperature,  through  the  worm  of  a 
calorimeter  at  such  a  rate  that  a  constant  difference  is  maintained 
between  the  temperature  of  the  entering  and  the  temperature  of 
the  escaping  gas.  Thus  the  number  of  calories  which  causes  a 
definite  thermal  change  in  a  certain  volume  of  the  gas  is  deter- 
mined, and  from  this  it  is  an  easy  matter  to  calculate  the  specific 
heat,  cp.  The  molecular  heat  at  constant  pressure  for  all  gases 
approaches  the  limiting  value,  6.5,  at  the  absolute  zero.  This 
relation,  due  to  Le  Chatelier,  may  be  expressed  thus, 
Mcp  =  6.5  +  aT, 

where  a  is  a  constant  for  each  gas.  The  value  of  a  for  hydrogen, 
oxygen,  nitrogen  and  carbon  monoxide  is  0.001,  for  ammonia, 
0.0071  and  for  carbon  dioxide,  0.0084.  As  the  complexity  of  the 
gas  increases  the  value  of  a  becomes  numerically  greater. 

The  experimental  determination  of  the  specific  heat  of  a  gas  at 
constant  volume  is  difficult  and  the  results  obtained  are  not 
trustworthy.  The  chief  cause  of  the  inaccuracy  of  the  results 

*  The  molecular  heat  of  a  gas  is  equal  to  the  product  of  its  specific  heat  and  its 
molecular  weight. 


96 


THEORETICAL  CHEMISTRY 


is  that  the  vessel  containing  the  gas  absorbs  so  much  more  heat 
than  the  gas  itself  that  the  correction  is  many  times  larger  than 
the  quantity  to  be  measured.  The  specific  heat  at  constant  vol- 
ume is  almost  always  obtained  by  indirect  methods,  as  for  example 
by  means  of  the  preceding  formula 

M  (cp  -  cv)  =  R  =  2  cal., 

in  which  the  values  of  M  and  cp  are  known. 

The  molecular  heats  of  some  of  the  commoner  gases  and  vapors 
are  given  in  the  subjoined  table  together  with  the  ratio   cp/cv. 

MOLECULAR  SPECIFIC  HEATS. 


•    Gas. 

Mcp 

Mcv 

cp/cv=y 

Argon 

66 

Helium 

66 

Mercury 

66 

Hydrogen 

6  88 

4.88 

.41 

Oxygen                               

6  96 

4.96 

.40 

Nitrogen  .                

6.93 

4.93 

.41 

Chlorine  

8.58 

6.58 

.30 

Bromine  

8.88 

6.88 

.29 

Nitric  oxide  

6.95 

4.95 

.40 

Carbon  monoxide 

6  86 

4.86 

41 

Hydrochloric  acid 

6  68 

4  68 

43 

Carbon  dioxide 

9  55 

7.55 

.26 

Nitrous  oxide         .      .             ... 

9.94 

7.94 

.25 

Water  

8.65 

6.65 

.28 

Sulphur  dioxide  

9.88 

7.88 

.25 

Ozone 

1  29 

Ether 

35  51 

33  51 

1  06 

The  Ratio  of  the  Two  Specific  Heats.  There  are  two  methods 
by  which  the  ratio  cp/cv  can  be  determined  directly,  one  due  to 
Clement  and  Desormes  *  and  the  other  due  to  Kundt.f 

Method  of  Clement  and  Desormes.  The  apparatus  devised  by 
these  investigators  consists,  as  is  shown  in  Fig.  21,  of  a  glass 
balloon  flask,  A,  of  about  20  liters  capacity,  furnished  with  two 
stop-cocks,  D  and  E,  and  a  manometer,  C.  The  stop-cock  D 
has  an  aperture  nearly  as  large  as  the  diameter  of  the  neck  of  the 

*  Jour,  de  phys.,  89,  321,  428  (1819). 

t  Pogg.  Ann.,  127,  497  (1866);   135,  337,  527  (1868). 


GASES 


97 


flask,  B.  To  determine  the  ratio  of  the  two  specific  heats,  the 
flask  is  filled  with  the  gas  under  a  pressure  slightly  greater  than 
barometric  pressure.  The  manometer  C  serves  to  measure  the 
pressure  of  the  gas  within  A.  After  the  value  of  the  pressure 
has  been  read  on  the  manometer,  the  stop-cock  D  is  opened 
momentarily  to  the  air,  thus  permitting  the  pressure  of  the  gas 
to  fall  adiabatically  to  that  of  the  atmosphere.  The  stop-cock 


Fig.  21. 

is  then  closed  and  the  flask  is  allowed  to  stand  for  a  few  moments 
until  its  contents,  which  has  cooled  by  adiabatic  expansion,  has 
regained  the  temperature  of  the  room.  The  pressure  on  the 
manometer  is  then  observed.  Let  the  initial  pressure  of  the  ga£ 
be  denoted  by  p0,  and  atmospheric  pressure  by  P.  If  the  initial 
and  final  specific  volumes  are  denoted  by  v0  and  vi,  then  for  an 
adiabatic  process,  we  have 

P 

Po 


98  THEORETICAL  CHEMISTRY 

The  value  of  the  final  specific  volume  is  determined  from  the 
final  pressure,  p\,  by  an  application  of  Boyle's  law,  the  pressure  p\ 
being  developed  isothermally. 
Thus, 


and  consequently 

L 

Po 
or 

=  log  P  -  log  po  t 
log  pi  —  log  po 

Method  of  Kundt.     According  to  the  formula  of  Laplace  for  the 
velocity  of  transmission  of  a  sound  wave  in  a  gas,  we  have 


in  which  p  and  d  denote  the  pressure  and  density  of  the  gas,  and 
7  is  the  ratio  of  the  two  specific  heats.  If  the  wave  velocities  in 
two  different  gases,  whose  densities  are  d\  and  d2  under  the  same 
conditions  of  temperature  and  pressure,  be  denoted  by  Vi  and  v2, 
we  may  write 

or  replacing  the  densities  of  these  gases  by  their  respective  molec- 
ular weights,  MI  and  M2,  we  have 


The  ratio  of  the  velocities  of  the  two  waves  can  be  measured  by 
means  of  the  apparatus  shown  in  Fig.  22.  A  wide  glass  tube 
about  li  meters  in  length  is  furnished  with  two  side  tubes,  E  and 
F.  Into  one  end  of  the  tube  is  inserted  the  glass  rod  BD  which 
is  clamped  at  its  middle  point  by  a  tightly  fitting  cork,  C.  The 
other  end  of  the  tube  is  closed  by  means  of  the  plunger  A.  A 
small  amount  of  lycopodium  powder  is  placed  upon  the  bottom  of 


GASES 


99 


the  tube  and  is  distributed  uniformly  by  gently  tapping  the  walls 
of  the  tube.  The  gas  in  which  the  velocity  of  the  sound  wave 
is  to  be  determined  is  introduced  into  the  tube  through  E,  and 
the  displaced  air  escapes  at  F.  When  the  tube  is  filled,  E  and  F 
are  closed  by  means  of  rubber  caps,  and  a  piece  of  moistened 
chamois  leather  is  drawn  along  BD  causing  it  to  vibrate  longitudi- 
nally and  to  emit  a  shrill  note.  The  vibrations  are  taken  up  by  the 


Fig.  22. 

gas  in  the  tube  and  the  powder  arranges  itself  in  a  series  of  heaps 
corresponding  to  the  nodes  of  vibration.  If  the  nodes  are  not 
sharply  defined,  then  A  should  be  moved  in  or  out  until  they 
become  so.  If  Xi  is  the  distance  between  two  heaps  or  nodes, 
then  2  Xi  will  be  the  wave  length  of  the  note  emitted  by  the  rod 
BD,  and  if  n  represents  the  number  of  vibrations  per  second  of 
the  note  emitted,  we  have  for  the  velocity  of  sound  in  the  gas 

vi  =  2  nXi. 

Similarly  if  a  second  gas  be  introduced  into  the  tube  we  shall 
have 


v2  =  2  n\2- 


Therefore, 


X2 


(2) 


Substituting  in  equation  (1),  we  have 

Xi 


or 


7i  =  72 


Mi 


(3) 


If  the  second  gas  is  air,  as  is  usually  the  case,  72  =  1.405  and  M2  = 
28.74,  (mol.  wt.  of  hydrogen  -4-  density  of  hydrogen  referred  to  air, 
or  2  -v-  0.0696  =  28.74)  or  equation  (3)  becomes 


100  THEORETICAL  CHEMISTRY 

\!2         Mi 

71  =   L4°5V'2874' 

Thus,  7  for  any  gas  can  be  determined  by  this  method  provided 
we  know  its  value  for  another  gas  of  known  molecular  weight. 

Specific  Heat  of  Gases  and  the  Kinetic  Theory.  In  terms  of 
the  kinetic  theory,  the  energy  of  a  gas  may  be  considered  to  be 
made  up  of  three  parts:  (1)  the  translational  energy  of  the  mole- 
cules, commonly  termed  their  kinetic  energy,  (2)  the  intramolec- 
ular kinetic  energy,  and  (3)  the  potential  energy  due  to  inter- 
atomic attraction  within  the  molecules.  When  a  gas  is  heated 
at  constant  volume  all  three  of  these  factors  of  the  total  energy 
of  the  molecule  may  be  affected.  It  is  fair  to  assume,  however, 
that  when  a  monatomic  gas,  such  as  mercury  vapor,  is  heated, 
all  of  the  heat  energy  supplied  is  used  to  augment  the  translational 
kinetic  energy  of  the  molecules.  As  we  have  seen,  the  fundamental 
kinetic  equation 

pv  =  1/3  nmc2 
may  be  written 

pv  =  2/3-l/2nmc2, 

and  since  1/2  nmc2  represents  the  total  kinetic  energy  of  the  gas, 
we  have 

pv  =  2/3  kinetic  energy  of  1  mol., 
or 

kinetic  energy  of  1  mol.  =  3/2  pv. 

But  pv  =  2T  calories,  therefore 

kinetic  energy  of  1  mol.  =  3  T  cal. 

The  kinetic  energy  of  a  constant  volume  of  any  gas  at  the  temper- 
atures TI  and  T2)  is  given  by  the  following  equations:  - 

3/2  ^0  =  3  T7!,  (1) 

and 

3/2^  =  3  TV  (2) 

Subtracting  (1)  from  (2)  we  obtain 

3/2  (p,  -Pl)v  =  3  (T2  -  7U  (3) 

and  for  an  increase  in  temperature  of  1°,  (3)  becomes 
3/2  (pa  -  Pl)  v  =  3  cal. 


GASES  ;  101 

The  molecular  kinetic  energy  of  one  mole  of  a  monatomic  gas 
at  constant  volume  is  thus  increased  by  3  calories  for  each  degree 
rise  in  temperature.  As  has  already  been  shown, 

M  (cp  —  cr)  =  2  cal., 
therefore,  since  Mcv  =  3  calories,  Mcp  =  3  +  2  =  5  calories,  and 

Mcp      5        aa 

T=TTF^  =  o=  L66- 
Mcv      3 

This  value  of  7  is  in  perfect  agreement  with  the  results  of  the 
experiments  on  mercury  vapor  which  is  known  to  be  monatomic. 
The  converse  of  this  method  has  been  employed  by  Ramsay  to 
prove  that  the  rare  gases  of  the  atmosphere  are  monatomic,  the 
value  of  7  for  all  of  these  gases  being  1.66.  In  the  case  of  poly- 
atomic molecules  the  heat  energy  supplied  is  not  only  used  in 
increasing  their  translational  kinetic  energy,  but  also  in  the 
performance  of  work  within  the  molecule.  The  value  of  the 
internal  work  is  indeterminate,  but  it  is  without  doubt  constant 
for  any  one  gas.  If  the  internal  work  be  represented  by  a,  then 
the  value  of  the  ratio  of  the  two  specific  heats  will  be 


Mcv      3  +  a 


Reference  to  the  table  on  p.  96,  giving  the  value  of  7  for  differ- 
ent gases,  will  show  that  this  deduction  from  the  kinetic  theory 
is  in  perfect  agreement  with  the  experimental  facts.  With 
increasing  complexity  of  the  molecule,  it  is  apparent  that  the 
amount  of  heat  expended  in  doing  internal  work  should  increase, 
and  therefore  the  specific  heat  should  increase  also.  Inspection 
of  the  table  confirms  this  deduction.  The  specific  heat  of  mona- 
tomic gases  is  independent  of  the  temperature  while  the  specific 
heat  of  polyatomic  gases  increases  slightly.  These  results  may 
justly  be  regarded  as  among  the  greatest  triumphs  of  the  kinetic 
leory  of  gases. 


102  THEORETICAL   CHEMISTRY 

PROBLEMS. 

1.  The  volume  of  a  quantity  of  gas  is  measured  when  the  barometer 
stands  at  72  cm.,  and  is  found  to  be  646  cc.:   what  would  its  volume  be 
at  normal  pressure?  Ans.   612  cc. 

2.  At  what  pressure  would  the  gas  in  the  preceding  problem  have  a 
volume  of  580  cc.?  Ans.   80.19  cm. 

3.  A  certain  quantity  of  oxygen  occupies  a  volume  of  300  cc.  at  0°: 
find  its  volume  at  91°.  Ans.  400  cc. 

4.  The  weight  of  a  liter  of  air  under  standard  conditions  is  1.293  grams: 
to  what  temperature  must  the  air  be  heated  so  that  it  may  weigh  exactly 
1  gram  per  liter?  Ans.   79°.99. 

5.  At  what  temperature  will  the  volume  of  a  given  mass  of  gas  be 
exactly  double  what  it  is  at  17°?  Ans.   307°. 

6.  On  heating  a  certain  quantity  of  mercuric  oxide  it  is  found  to  give 
off  380  cc.  of  oxygen,,  the  temperature  being  23°  and  the  barometric 
height  74  cm.;    what  would  be  the  volume  of  the  gas  under  standard 
conditions?  Ans.   341.25  cc. 

7.  A  liter  of  air  weighs  1.293  grams  under  standard  conditions.     At 
what  temperature  will  a  liter  of  air  weigh  1  gram,  the  pressure  being  72  cm.? 

Ans.   61°.43. 

8.  A  quantity  of  air  at  atmospheric  pressure  and  at  a  temperature  of 
7°  is  compressed  until  its  volume  is  reduced  to  one-seventh,  the  temper- 
ature rising  20°  during  the  process:   find  the  pressure  at  the  end  of  the 
operation.  Ans.   7.3  atmos. 

9.  The  weight  of  a  liter  of  nitrogen  under  standard  conditions  is  1.2579 
grams.    Calculate  the  specific  gas  constant,  r.          Ans.  3007  gr.  cm. 

10.  The  time  of  outflow  of  a  gas  is  21.4  minutes,  the  corresponding 
time  for  hydrogen  is  5.6  minutes.     Find  the  molecular  weight  of  the  gas. 

Ans.   29.2. 

11.  Calculate  the  molecular  weight  of  chloroform  from  the  following 
data:  — 

Weight  of  chloroform  taken   0 .220  gr. 

Volume  of  air  collected  over  water   45 .0  cc. 

Temperature  of  air 20° 

Barometric  pressure  755 .0  mm. 

Pressure  of  aqueous  vapor  at  20° 17 .4  mm. 

Ans.  121.1. 


GASES  103 

12.  The  density  of  a  gas  is  0.23  referred  to  mercury  vapor.    What  is 
its  molecular  weight?  Ans.  46. 

13.  Phosphorus  pentachloride  dissociates  according  to  the  equation 


The  molecular  weight  of  PC15  is  208.28.    At  182°  the  density  is  73.5  and 
at  230°  it  is  62.     Find  the  degree  of  dissociation  at  the  two  temperatures. 

Ans.   am°  =  0.417,  cw  =  0.68. 

14.  The  specific  heat  at  constant  volume  for  argon  is  0.075,  and  its 
molecular  weight  is  40.     How  many  atoms  are  there  in  the  molecule? 

Ans.    1. 

15.  What  is  the  specific  heat  of  carbon  dioxide  at  constant  volume,  its 
molecular  weight  being  44  and  the  temperature  50°.  Ans.   0.164. 

16.  The  specific  heat  for  constant  pressure  of  benzene  is  0.295  :  what  is 
the  specific  heat  for  constant  volume?  Ans.  0.27. 


CHAPTER  VII. 
LIQUIDS. 

General  Characteristics  of  Liquids.  The  most  marked  char- 
acteristic of  the  liquid  state  is  that  a  given  mass  of  liquid  has  a 
definite  volume  but  no  definite  form.  The  volume  of  a  liquid  is 
dependent  upon  temperature  and  pressure  but  to  a  much  smaller 
degree  than  is  the  volume  of  a  gas.  The  formulas  in  which  the  vol- 
ume of  a  liquid  is  expressed  as  a  function  of  temperature  and  pres- 
sure are  largely  empirical,  and  contain  constants  dependent  upon 
the  nature  of  the  liquid.  This  is  undoubtedly  due  to  the  fact  that 
in  the  liquid  state  the  molecules  are  much  less  mobile  than  in  the 
gaseous  state.  The  distance  between-  contiguous  molecules  being 
much  less  in  liquids  than  in  gases,  the  mutual  attraction  is  increased 
while  the  mobility  is  correspondingly  diminished.  That  liquids 
represent  a  more  condensed  form  of  matter  than  gases  is  shown 
by  the  change  in  volume  which  results  when  a  liquid  is  vaporized : 
thus,  1  cc.  of  water  at  the  boiling  point  when  vaporized  at  the  same 
temperature  occupies  a  volume  of  about  1700  cc.  A  liquid  con- 
tains less  energy  than  a  gas,  since  energy  is  always  required  to 
transform  it  into  the  gaseous  state.  Since  gases  can  be  liquefied 
by  increasing  the  pressure  and  lowering  the  temperature,  and  since 
liquids  can  be  vaporized  by  lowering  the  pressure  and  increasing 
the  temperature,  it  is  apparent  that  there  is  no  generic  difference 
between  the  two  states  of  matter. 

Connection  Between  the  Gaseous  and  Liquid  States.  If  a  gas 
is  compressed  isothermally,  its  state  may  change  in  either  of  two 
ways  depending  upon  the  temperature:  —  (1)  The  volume  at  first 
diminishes  more  rapidly  than  the  pressure  increases,  then  in  the 
same  ratio  and  lastly  more  slowly.  When  the  pressure  attains  a 
very  high  value  the  volume  is  but  slightly  altered.  This  case 
has  already  been  considered  in  the  preceding  chapter.  (2)  The 
volume  changes  more  rapidly  than  the  pressure  until,  when  a  cer- 

104 


LIQUIDS  105 

tain  pressure  is  reached,  the  gas  ceases  to  be  homogeneous,  partial 
liquefaction  resulting.  For  a  constant  temperature,  the  pressure  at 
which  liquefaction  occurs  is  invariable  for  a  given  gas,  while  the  vol- 
ume steadily  diminishes  until  liquefaction  is  complete.  Only  when 
the  whole  mass  of  gas  has  been  liquefied  is  it  possible  to  increase 
the  pressure  and  then,  owing  to  the  small  compressibility 'of  liquids, 
a  large  increase  in  pressure  is  required  to  produce  a  slight  dimin- 
ution in  volume.  If  the  temperature  is  above  a  certain  point, 
dependent  upon  the  nature  of  the  gas,  the  phenomena  of  com- 
pression will  follow  (1);  if  below  this  point,  the  process  will  follow 
(2).  That  a  gas  may  behave  in  either  of  the  above  ways  was 
first  clearly  recognized  by  Andrews  *  in  1869,  in  connection  with 
his  experiments  on  the  liquefaction  of  carbon  dioxide.  He  found 
that  if  carbon  dioxide  was  compressed,  keeping  the  temperature 
at  0°,  the  volume  changes  more  rapidly  than  the  pressure,  lique- 
faction resulting  when  a  pressure  of  35.4  atmospheres  was  reached. 
As  the  temperature  was  raised,  he  found  that  a  higher  pressure 
was  required  to  liquefy  the  gas,  until  at  temperatures  above 
30°. 92  it  was  no  longer  possible  to  condense  the  gas  to  the  liquid 
state.  The  temperature  above  which  it  was  no  longer  possible 
to  liquefy  the  gas  he  termed  the  critical  temperature.  In  like 
manner  the  pressure  required  to  liquefy  the  gas  at  the  critical 
temperature,  he  termed  the  critical  pressure,  and  the  volume 
occupied  by  the  gas  or  the  liquid  under  these  conditions  he  called 
the  critical  volume. 

Isothermals  of  Carbon  Dioxide.  The  results  of  Andrew's 
experiments  f  on  the  liquefaction  of  carbon  dioxide  are  shown  in 
Fig.  23,  in  which  the  ordinates  represent  pressures  and  the  abscissae 
the  corresponding  volumes  at  constant  temperature.  The  curves 
obtained  by  plotting  volumes  against  pressures  at  constant 
temperatures  are  called  isothermals.  For  a  gas  which  follows 
Boyle's  law,  the  isothermals  will  be  a  series  of  equilateral  hy- 
perbolas. This  condition  is  approximately  fulfilled  by  air,  for 
which  three  isothermals  are  given  in  the  diagram.  At  48°.  1  the 
isothermal  for  carbon  dioxide  is  nearly  hyperbolic,  but  as  the 

*  Trans.  Roy.  Soc.  159,  583  (1869). 
f  loc.  cit. 


106 


THEORETICAL  CHEMISTRY 


temperature  becomes  lower,  the  isothermals  deviate  more  and  more 
from  those  for  an  ideal  gas.     At  the  critical  temperature,  30°. 92, 


Carbon  Dioxide  — 


Air 


Volume 
Fig.  23. 

the  curve  is  almost  horizontal  for  a  short  distance,  showing  that  for 
a  very  slight  change  in  pressure  there  is  an  enormous  shrinkage 
in  volume.  At  still  lower  temperatures,  21°.  1  and  13°.  1,  the 


LIQUIDS  107 

horizontal  portions  of  the  curves  are  much  more  pronounced, 
indicating  that  during  liquefaction  there  is  no  change  in  pressure. 
When  liquefaction  is  complete  the  curves  rise  abruptly,  showing 
that  the  change  in  volume  is  extremely  small  for  a  large  increase 
in  pressure;  in  other  words  the  liquefied  gas  possesses  a  small 
coefficient  of  compressibility.  At  any  point  within  the  parabolic 
area,  indicated  by  the  dotted  line  ABC,  both  vapor  and  liquid  are 
coexistent  ;  at  any  point  outside,  only  one  form  of  matter,  either 
liquid  or  vapor,  is  present.  Andrew's  experiments  show  that 
there  is  no  fundamental  difference  between  a  gas  and  a  liquid. 
It  is  apparent  from  the  diagram  that  when  carbon  dioxide  is  sub- 
jected to  great  pressures  above  its  critical  temperature  it  behaves 
more  like  a  liquid  than  a  gas,  in  fact  it  is  difficult  to  determine 
whether  a  highly  compressed  gas  above  its  critical  temperature 
should  be  classified  as  a  gas  or  as  a  liquid. 

Van  der  Waals'  Equation  and  the  Continuity  of  the  Gaseous 
and  Liquid  States.  In  the  preceding  chapter  we  have  learned 
that  the  fundamental  gas  equation 

pv  =  RT 

is  only  strictly  applicable  to  an  ideal  gas,  and  that  the]  behavior 
of  actual  gases  is  represented  with  considerable  accuracy,  even  at 
high  pressures,  by  the  equation  of  van  der  Waals, 


If  this  equation  be  arranged  in  descending  powers  of  v,  we  have 

,/,    .  RT\  .      a      ab 
v3  -  v2  6H  ---  )+v  ---  =  0.  (1) 

V       p  I      p     p 

This  being  a  cubic  equation  has  three  possible  solutions,  each  val- 
ue of  p  affording  three  corresponding  values  of  v;  a,b,R  and  T  being 
treated  as  constants.  The  three  roots  of  this  equation  are  either 
all  real,  or  one  is  real  arid  two  are  imaginary,  depending  upon  the 
values  of  the  constants.  That  is  to  say,  at  one  temperature 
and  pressure  the  values  of  a  and  b  may  be  such,  that  v  has  three 
real  values,  while  at  another  temperature  and  pressure,  v  may 
have  one  real  and  two  imaginary  values.  In  the  accompanying 


108 


THEORETICAL  CHEMISTRY 


diagram,  Fig.  22,  a  series  of  graphs  of  the  equation  for  different 
values  of  T  is  given.  It  will  be  observed  that  these  curves  bear 
a  striking  resemblance  to  the  isotherms  of  carbon  dioxide  estab- 
lished by  the  experiments  of  Andrews.  In  the  case  of  the  theo- 


Volume 
Fig.  24. 

retical  curves  there  are  no  sudden  breaks  such  as  appear  in  the 
actual  discontinuous  passage  from  the  gaseous  to  the  liquid  state. 
Instead  of  passing  from  B  to  D  along  the  wavelike  path  BaCbD, 
experiment  has  shown  that  the  substance  passes  directly  from  the 
state  B  to  the  state  D  along  the  straight  line  BD.  It  is  here 


LIQUIDS  ,  109 

that  van  der  Waals'  equation  fails  to  apply.  As  has  been  pointed 
out  the  substance  between  these  two  points  is  not  homogeneous, 
being  partly  gaseous  and  partly  liquid.  Attempts  have  been 
made  to  realize  the  portion  of  the  curve  BaCbD  experimentally. 
By  studying  supersaturated  vapors  and  superheated  liquids  it 
has  been  found  possible  to  follow  the  theoretical  curve  for  short 
distances  between  B  and  D  without  discontinuity,  but  owing  to 
the  instability  of  the  substance  in  this  region,  it  is  evident  that 
the  complete  isothermal  and  continuous  transformation  of  a  gas 
into  a  liquid  can  never  be  effected.  Van  der  Waals  has  called 
attention  to  the  fact  that  in  the  surface  layer  of  a  liquid,  where 
unique  conditions  prevail,  it  is  quite  possible  that  such  unstable 
states  may  exist,  and  that  there  the  transition  from  liquid  to  gas 
may  in  reality  be  a  continuous  process.  The  diagram  shows  that 
as  T  increases,  the  wave-like  portion  of  the  isothermals  becomes 
less  pronounced  and  eventually  disappears,  when  the  points  5, 
C  and  D  coalesce.  At  this  point  the  three  roots  of  the  equa- 
tion become  equal,  the  volume  of  the  liquid  becoming  identical 
with  the  volume  of  the  gas.  The  substance  at  this  point  is  in  the 
critical  condition.  Since  under  these  conditions  the  three  roots 
of  the  equation 

,      I,    .  RT\      .    a        ab      _  „. 

Vs  -  (b-\ }v2+  -v =  0  (1) 

V          p  I         P         P 

are  equal,  we  may  write  vi  =  v2  =  v3  =  vc,  the  subscript  c  indicat- 
ing the  critical  state.     Then  equation  (1)  must  be  equivalent  to 
(v  -  vc)s  =  v*-3 vcv*  +  3 v*v  -  v*  =  0.  (2) 

Equating  the  corresponding  coefficients  of  equations  (1)  and  (2), 
we  have 

:;:r*       ;;; 

PC 

and  ,/  =  ^.  (5) 

PC 

Dividing  equation  (5)  by  equation  (4),  we  have 

vc  =  3b,  (6) 


110  THEORETICAL  CHEMISTRY 

and  substituting  this  value  in  equation  (4),  we  obtain 


Lastly,  substituting  the  values  of  vc  and  pc}  given  in  equations 
(6)  and  (7),  in  equation  (3),  we  have 

T         8  a 

-i  c   — 


27  bR 

Therefore, 

rn       =  Q 

Or  expressing  the  constants  a,  b  and  R  in  terms  of  the  critical 
values  of  pressure,  temperature  and  volume,  we  have 


(10) 


and 


By  means  of  equations  (6),  (7)  and  (8)  it  is  possible  to  calculate 
the  critical  constants  of  a  gas  when  the  constants  a  and  b  of  van 
der  Waals'  equation  are  known.  If  we  take  carbon  dioxide  as  an 
example,  for  which  a  =  0.00874  and  b  =  0.0023  we  obtain 
vc  =  0.0069  (observed  value  =  0.0066),  pc  =  61  atmospheres, 
(observed  value  =  70  atmospheres),  Tc  =  305°.5  abs.,  (observed 
value  303°.  9  abs.).  Conversely  by  means  of  equations  (10)  and 
(11)  the  value  of  a  and  b  can  be  calculated  when  the  critical  data 
are  given. 

Corresponding  Conditions.  If  in  the  equation  of  van  der 
Waals,  the  values  of  p,  v  and  T  be  expressed  as  fractions  of  the 
corresponding  critical  values,  we  may  write 

p  =  apc, 
v  =  (3vc 
and 

T  =  iTc. 


LIQUIDS  111 

Substituting  these  values  in  the  equation 


we  have 

(apc  +  (0ve  -  6) 


and  replacing  pc,  vc,  and  Tc  by  their  values  given  in  equations  (6), 
(7)  and  (8)  of  the  preceding  paragraph,  we  obtain 


which  is  van  der  Waals'  reduced  equation  of  condition. 

In  this  equation  everything  connected  with  the  individual 
nature  of  the  substance  has  vanished,  thus  making  it  applicable 
to  all  substances  in  the  liquid  or  gaseous  state  in  the  same  way 
that  the  fundamental  gas  equation  is  applicable  to  all  gases  irre- 
spective of  their  specific  nature.  It  has  been  shown,  however, 
that  the  equation  is  not  entirely  trustworthy  and  at  best  can  be 
considered  as  little  more  than  a  rough  approximation.  The 
chief  point  to  be  observed  in  connection  with  this  equation  is 
that  whereas  for  gases,  the  corresponding  values  of  temperature, 
pressure  and  volume,  measured  in  the  ordinary  units,  may  be 
compared,  it  is  necessary  in  the  case  of  liquids  to  make  the  com- 
parison under  corresponding  conditions.  For  example,  the  molec- 
ular volumes  of  two  liquids  are  to  be  compared,  not  at  room 
temperature  but  at  temperatures  which  are  equal  fractions  of 
their  respective  critical  temperatures.  Such  temperatures  van 
der  Waals  called  corresponding  temperatures. 

By  way  of  illustration,  suppose  we  wish  to  compare  alcohol 
and  ether  with  respect  to  some  particular  property,  such  as 
surface  tension.  If  the  surface  tension  of  alcohol  be  measured 
at  60°,  at  what  temperature  must  a  similar  measurement  be 
made  with  ether  in  order  that  the  results  may  be  comparable? 
The  critical  temperature  of  alcohol  is  243°  C.  or  516°  absolute; 
that  of  ether  is  194°  C.  or  467°  absolute.  Then  according  to  van 
der  Waals'  definition  of  corresponding  conditions,  the  temperature, 


112 


THEORETICAL  CHEMISTRY 


t,  at  which  measurements  should  be  made  with  ether  will  be  given 
by  the  proportion,  273  +  t  :  467  ::  273  +  60  :  516,  or  t  =  28°  C. 
By  making  comparisons  of  various  properties  at  corresponding 
temperatures  it  has  been  found  that  greater  regularities  are 
observed  than  when  comparisons  are  made  at  the  same  tempera- 
ture, thus  justifying  the  claim  of  van  der  Waals. 

Liquefaction  of  Gases.  The  history  of  the  liquefaction  of 
gases  has  for  a  long  time  been  regarded  as  one  of  the  most 
interesting  chapters  of  physical  science.  Among  the  first  success- 


Fig.  25. 

ful  workers  in  this  field  was  Faraday.*  He  liquefied  practically 
all  of  the  gases  which  condense  under  moderate  pressures  and  at 
not  very  low  temperatures.  A  sketch  of  the  apparatus  used  by 
Faraday  is  shown  in  Fig.  25.  It  consisted  of  an  inverted 
V-shaped  tube,  in  one  end  of  which  was  placed  some  solid  which 
would  liberate  the  desired  gas  on  heating,  while  the  other  end 
was  sealed  and  immersed  in  a  freezing  mixture.  When  the  sub- 
stance at  A  had  been  heated  long  enough  to  liberate  considerable 
gas,  the  pressure  within  the  tube  became  sufficiently  high  to  cause 
the  gas  to  liquefy  at  the  temperature  of  the  end  B.  Thus  chlorine 
*  Phil.  Trans.  113,  189  (1823). 


LIQUIDS  113 

hydrate  was  heated  in  the  tube  and  the  liberated  chlorine  was 
condensed  at  B  as  a  yellow  liquid.  In  1834,  Thilorier  *  succeeded 
in  liquefying  carbon  dioxide  in  quite  large  amounts  by  the  use  of 
a  new  form  of  apparatus.  In  connection  with  his  experiments 
on  liquid  carbon  dioxide,  he  observed  that  when  it  was  allowed  to 
vaporize,  enough  heat  was  absorbed  to  lower  the  temperature 
below  its  freezing  point,  solid  carbon  dioxide  being  obtained.  He 
discovered  that  a  mixture  of  solid  carbon  dioxide  and  ether  was  a 
powerful  refrigerant,  and  that  under  diminished  pressure  the 
mixture  gave  temperatures  ranging  from  —  100°  C.  to  —  110°  C. 
This  mixture  is  known  today  as  Thilorier' s  mixture.  Faraday  f 
undertook  the  liquefaction  of  the  so-called  permanent  gases  in  1845. 
In  this  second  series  of  experiments  by  Faraday  he  employed 
higher  pressures  than  in  his  earlier  experiments,  and  also  made 
use  of  the  newly  discovered  Thilorier  mixture  as  a  refrigerant. 
He  was  partially  successful  in  his  attempt  to  liquefy  the  hitherto 
noncondensible  gases.  He  liquefied  ethylene,  phosphine  and 
hydrobromic  acid  and  also  solidified  ammonia,  cyanogen,  and 
nitrous  oxide.  He  failed  to  liquefy  hydrogen,  oxygen,  nitrogen, 
nitric  oxide  and  carbon  monoxide.  No  further  advance  in  the 
liquefaction  of  gases  was  made  until  the  year  1869  when  Andrews 
pointed  out  the  importance  of  cooling  the  gas  below  its  critical 
temperature.  This  discovery  explained  why  so  many  of  the 
earlier  experiments  had  failed,  and  opened  the  way  to  the  brilliant 
successes  of  the  latter  part  of  the  nineteenth  century.  In  1877, 
Cailletet  f  and  Pictet,  §  working  independently,  succeeded  in 
liquefying  oxygen.  Cailletet  subjected  the  gas  to  a  pressure  of 
about  300  atmospheres  using  boiling  sulphur  dioxide  as  a  refriger- 
ant. The  gas  was  further  cooled  by  suddenly  releasing  the  pres- 
sure and  allowing  it  to  expand.  In  addition  to  oxygen  he  also 
liquefied  air,  nitrogen  and  possibly  hydrogen.  Shortly  afterward 
in  1883,  the  Polish  scientists,  Wroblewski  and  Olszewski,  1F 

*  Lieb.  Ann.,  30,  122  (1839). 
t  Phil.  Trans.,  135,  155  (1845). 
t  Compt.  rend.,  85,  1217  (1877). 
§  Ibid.,  85,  1214,  1220  (1877). 
1  Wied.  Ann.,  20,  243  (1883). 


114  THEORETICAL  CHEMISTRY 

lished  an  account  of  their  interesting  and  highly  important  work. 
In  their  experiments  they  subjected  the  gas  to  be  liquefied  to  high 
pressure,  and  simultaneously  cooled  it  to  a  very  low  temperature. 
Among  the  refrigerants  used  by  them  was  liquid  ethylene,  which 
was  allowed  to  boil  off  under  diminished  pressure,  giving  a  temper- 
ature of  -  130°  C.  At  this  temperature,  a  pressure  of  only 
20  atmospheres  was  sufficient  to  condense  oxygen  to  the  liquid 
state.  Having  liquefied  oxygen,  nitrogen,  air  and  carbon  mon- 
oxide, and  having  determined  the  boiling-points  of  these  gases 
under  atmospheric  pressure,  they  proceeded  to  use  these  liquefied 
gases  as  refrigerants,  allowing  them  to  boil  off  under  diminished 
pressure,  thus  obtaining  temperatures  as  low  as  —  200°  C.  A 
very  small  amount  of  liquid  hydrogen  was  obtained  in  this  way. 
Subsequent  attempts  by  these  same  experimenters  to  liquefy 
hydrogen,  while  not  much  more  successful  than  their  former 
attempts,  enabled  them  to  determine  its  boiling-point.  Shortly 
after  the  publication  of  the  first  papers  of  Wroblewski  and 
Olszewski,  Dewar  *  devised  a  new  form  of  apparatus  for  lique- 
fying air,  oxygen  and  nitrogen  on  a  comparatively  large  scale. 
He  also  introduced  the  well-known  vacuum-jacketed  flasks  and 
tubes  which  greatly  facilitated  carrying  out  experiments  with 
liquefied  gases.  In  1895,  Linde  in  Germany  and  Hampson  in 
England  simultaneously  and  independently  constructed  machines 
for  the  liquefaction  of  air  in  large  quantities. 

In  the  method  devised  by  these  experimenters  the  air  is  not 
subjected  to  a  preliminary  cooling,  produced  by  the  rapid  evapora- 
tion of  a  liquefied  gas  under  diminished  pressure,  as  in  the  methods 
of  Wroblewski  and  Olszewski.  In  the  Linde  liquefier,  the  air  is 
compressed  to  about  200  atmospheres.  It  is  then  passed  through 
a  chamber  containing  anhydrous  calcium  chloride  to  remove  the 
greater  part  of  the  moisture,  after  which  it  is  cooled  by  allowing  it 
to  circulate  through  a  coiled  pipe  immersed  in  a  freezing  mixture. 
Nearly  all  of  the  moisture  remaining  in  the  air  is  deposited  on  the 
walls  of  the  pipe  in  the  form  of  frost.  The  air  then  enters  a  long 
spiral  tube  jacketed  with  a  non-conducting  material,  and  is  there 
allowed  to  expand  to  a  pressure  of  about  15  atmospheres. 

*  Proc.  Roy.  Inst,  1886,  550. 


LIQUIDS  115 

During  this  expansion  the  temperature  of  the  air  is  appreciably 
lowered.  When  the  air  has  traversed  the  spiral  tube,  it  is  still 
further  cooled  by  allowing  it  to  expand  to  a  pressure  equal  to  that 
of  the  atmosphere.  The  air  which  has  been  thus  cooled  is  then 
passed  backward  through  the  annular  space  between  the  spiral  tube 
and  a  concentric  jacket,  thus  cooling  the  entering  portion  of  air. 
Consequently  this  next  portion  of  air  expands  from  a  lower  initial 
temperature,  and  the  cooling  effect  is  increased.  In  like  manner, 
when  this  cooler  air  passes  backward  it  cools  still  further  the  next 
succeeding  portion,  and  eventually  the  temperature  is  reduced 
sufficiently  to  cause  a  small  amount  of  the  air  to  liquefy  as  it 
issues  from  the  end  of  the  spiral  tube.  The  remaining  portion  of 
the  air  which  has  not  been  liquefied,  passes  backward  through  the 
annular  tube  and  cools  the  following  portion  to  a  still  greater 
extent,  causing  a  larger  proportion  to  liquefy  on  expansion.  With 
a  3-horse-power  machine,  a  continuous  supply  of  0.9  liter  per  hour 
can  be  obtained.  Further  improvements  in  this  apparatus  have 
been  made  by  Dewar  and  Hampson,  and  by  means  of  it  Dewar's 
brilliant  successes  in  the  liquefaction  of  gases  have  been  achieved. 
The  most  efficient  apparatus  for  the  liquefaction  of  air  and  other 
gases  is  that  developed  by  Claude.*  The  essential  features  of 
his  liquefier  are  shown  in  Fig.  26.  The  air  is  first  compressed  to 
40  atmospheres  pressure  by  means  of  an  ordinary  compression 
pump  not  shown  in  the  diagram,  the  moisture  and  carbon  dioxide 
being  removed  as  in  Linde's  method.  It  then  enters  the  tube  A, 
which  in  reality  is  of  a  spiral  form,  and  divides  at  B.  A  portion 
enters  the  cylinder  D  through  a  valve  chest  similar  to  that  in  a 
steam  engine  forces  out  the  piston  and  causes  the  wheel,  TF, 
to  revolve,  thereby  doing  work  and  cooling  the  air.  The  cooled 
air  escapes  from  the  valve  chest  and  circulates  through  the  lique- 
fying chamber  L,  where  it  causes  the  portion  of  compressed  air 
entering  at  B  to  liquefy.  It  then  issues  from  the  liquefier  and 
traverses  M,  cooling  the  entering  portion  of  air  in  A,  and  finally 
returns  to  the  compressor.  The  pressure  of  the  air  when  it  issues 
from  D  is  almost  atmospheric,  and  its  temperature  is  below 
—  140°  C.  About  twenty-five  per  cent  of  the  power  consumed 
*  Compt.  rend.,  II,  500  (1900);  I,  1568  (1902);  II,  762,  823,  (1905). 


116 


THEORETICAL  CHEMISTRY 


in  compression  is  regained  by  the  motor.  The  apparatus  pro- 
duces about  1  liter  of  liquid  air  per  horse-power  hour.  By  means 
of  this  improved  apparatus,  based  upon  the  regenerative  principle, 
all  known  gases  have  now  been  liquefied,  the  last  to  succumb  being 


Liquid  Air 
40  atmos..— 140 


Fig.  26. 


helium  which  was  liquefied  in  1908,  by  Kammerlingh  Onnes  in  the 
Leyden  cryogenic  laboratory.  The  subjoined  table  gives  the 
critical  data  together  with  the  boiling  and  freezing  temperatures 
of  some  of  the  more  common  gases. 

CRITICAL  DATA  FOR  GASES. 


Gas. 

Crit.  Temp. 

Crit.  Press. 
(Atmos.) 

Boiling  Point 
(at  760  mm.). 

Freezing 
Point. 

Helium  

-267° 

Hydrogen  

-242° 

20 

-252°  5 

-258°.  9 

Air  

-140° 

39 

-191° 

Nitrogen  

-146° 

35 

-195°  5 

-210°.  5 

Oxygen 

-118° 

50  8 

—  182°  8 

-227° 

Carbon  monoxide  
Nitric  oxide 

-141° 
-  96° 

36 
64 

-190° 
—  153°  6 

-207° 
—  167° 

Carbon  dioxide  

+  31° 

73 

—  78° 

-  65° 

Hydrochloric  acid  
Ammonia        

+  51°.  3 
+130° 

81.5 
115 

-  35° 
—  33°  7 

-116° 

-  77° 

LIQUIDS  117 

Vapor  Pressure  of  Liquids.  According  to  the  kinetic  theory 
there  is  a  continuous  flight  of  particles  of  vapor  from  the  surface 
of  a  liquid  into  the  free  space  above  it.  At  the  same  time  the 
reverse  process  of  condensation  of  vapor  particles  at  the  surface 
of  the  liquid  is  taking  place.  Eventually  a  condition  of  equilib- 
rium will  be  established  between  the  liquid  and  its  vapor,  when 
the  rate  of  escape  will  be  exactly  counterbalanced  by  the  rate  of 
condensation  of  vapor  particles.  The  pressure  exerted  by  the 
vapor  of  a  liquid  when  equilibrium  has  been  attained  is  known  as 
its  vapor  pressure.  The  equilibrium  between  a  liquid  and  its  vapor 
is  dependent  upon  the  temperature.  For  every  temperature 
below  the  critical  temperature,  there  is  a  certain  pressure  at  which 
vapor  and  liquid  may  exist  in  equilibrium  in  all  proportions;  and 
conversely  for  every  pressure  below  the  critical  pressure,  there  is 
a  certain  temperature  at  which  vapor  and  liquid  may  exist  in 
equilibrium  in  all  proportions.  This  latter  temperature  is  termed 
the  boiling-point  of  the  liquid.  'The  vapor  pressure  of  a  liquid 
may  be  measured  directly  by  placing  a  portion  of  it  above  the 
mercury  in  the  vacuum  of  a  barometer  tube,  heating  to  the  desired 
temperature,  and  observing  the  depression  of  the  mercury  column. 
This  is  known  as  the  static  method.  It  is  open  to  the  objection 
that  the  presence  of  volatile  impurities  in  the  liquid  causes  too 
great  depression  of  the  mercury  column,  the  vapor  pressure  of 
the  impurity  adding  itself  to  that  of  the  liquid  whose  vapor 
pressure  is  sought.  A  better  method  for  the  measurement  of 
vapor  pressure  is  that  known  as  the  dynamic  method.  In  this 
method  the  pressure  is  maintained  constant  and  the  boiling 
temperature  is  determined  with  an  accurate  thermometer.  The 
boiling  temperatures  corresponding  to  various  pressures  may  be 
measured,  provided  we  have  a  suitable  device  for  changing  and 
measuring  the  pressure.  The  results  obtained  by  the  static  and 
dynamic  methods  agree  closely  if  the  liquid  is  pure,  but  if  volatile 
impurities  are  present  the  results  obtained  by  the  dynamic  method 
are  more  trustworthy.  A  method  for  the  measurement  of  vapor 
pressure  due  to  James  Walker  *  is  of  considerable  interest.  In 
this  method,  a  current  of  pure  dry  air  is  bubbled  through  a  weighed 
*  Zeit.  phys.  Chem.,  2,  602  (1888). 


118  THEORETICAL  CHEMISTRY 

amount  of  the  liquid  whose  vapor  pressure  is  to  be  determined. 
The  liquid  is  maintained  at  constant  temperature  and  its  loss  in 
weight  is  observed.  In  passing  through  the  liquid  the  air  will 
absorb  an  amount  of  vapor  directly  proportional  to  the  vapor 
pressure  of  the  liquid.  If  1  mol  of  liquid  is  absorbed  by  v  liters 

of  air,  then  we  have 

pv  =  RT, 

where  p  is  the  vapor  pressure  of  the  liquid,  and  T  its  temperature. 
If  vi  is  the  volume  of  air  which  absorbs  g  grams  of  the  vapor  of 
the  liquid  whose  molecular  weight  is  M,  then 

m  =  ± 
or 


In  this  equation,  Vi  denotes  the  total  volume  containing  g  grams 
of  the  liquid  in  the  form  of  vapor,  or  in  other  words  it  represents 
the  air  and  vapor  together.  Since  the  volume  of  the  air  is  in 
general  so  much  greater  than  that  of  the  vapor,  Vi  may  be  taken 
as  that  of  the  air  alone. 

Heat  of  Vaporization.  In  order  to  transform  a  liquid  into  a 
vapor  a  large  amount  of  heat  is  required.  Thus,  when  a  liquid 
is  heated  to  the  boiling-point,  the  volume  must  be  increased  against 
the  pressure  of  the  atmosphere,  external  work  being  done,  and 
when  the  boiling  temperature  is  reached  the  liquid  must  be  vapor- 
ized; the  heat  expended  in  causing  the  change  of  physical  state 
being  much  greater  than  that  required  to  expand  the  liquid.  An 
interesting  relation  between  the  heat  of  vaporization  and  the 
absolute  boiling-point  of  a  liquid  was  discovered  by  Trouton.* 
If  T  denotes  the  absolute  boiling-point  and  w  the  heat  of  vapor- 
ization of  1  gram  of  liquid  whose  molecular  weight  is  M,  then 
according  to  Trouton 

Mw_ 

jr       ~   ^1> 

or  in  words,  the  ratio  of  the  molecular  heat  of  vaporization  to  the 
absolute  boiling  temperature  of  a  liquid  is  constant,  the  numerical 

*  Phil.  Mag.  (5),  18,  54  (1884). 


LIQUIDS 


119 


value  of  the  ratio  being  approximately  21.  This  is  known  as 
Trouton's  law.  While  this  relation  holds  quite  well  for  many 
liquids,  Nernst  has  pointed  out  that  the  constant  varies  with  the 
temperature,  and  has  proposed  two  other  forms  of  the  equation. 
Bingham  has  simplified  the  equations  of  Nernst  to  the  following 
form  :  — 

17+0.011  T. 


While  this  modification  of  the  Trouton  equation  has  been  found 
to  hold  for  a  large  number  of  substances,  there  are  other  substances 
for  which  the  left  side  of  the  equation  has  a  value  greater  than 
that  of  the  right  side.  Bingham  infers  that  where  this  occurs,  the 
substance  in  the  liquid  state  has  a  greater  molecular  weight 
than  it  has  in  the  gaseous  state,  or  in  other  words,  the  liquid  is 
associated.  It  is  evident  that  an  associated  liquid  will  require 
an  expenditure  of  energy  over  and  above  that  required  for  vapori- 
zation, to  break  down  the  molecular  complex.  The  difference 
between  the  values  of  the  two  sides  of  the  equation  may  be 
taken  as  a  rough  measure  of  the  degree  of  association. 

Boiling-Point  and  Critical  Temperature.  An  interesting  rela- 
tion has  been  pointed  out  by  Guldberg  *  and  Guye.f  These 
two  investigators  have  shown  that  the  absolute  boiling  temper- 
ature of  a  liquid  is  about  two-thirds  of  its  critical  temperature. 
That  this  empirical  relation  holds  for  a  variety  of  different  sub- 
stances is  shown  in  the  accompanying  table. 

RELATION   OF  BOILING-POINT  TO  CRITICAL  TEMPERATURE. 


Substance. 

Tb 

Te 

Tb/Te 

Oxygen                 

90° 

155° 

0.58 

Chlorine  .         

240° 

414° 

0.58 

Sulphur  dioxide  
Ethyl  ether  
Ethyl  alcohol  

263° 
308° 
351° 

429° 
467° 
516° 

0.61 
0.66 
0.68 

Benzene  

353° 

562° 

0.63 

Water  

373° 

637° 

0.59 

Phenol  

454° 

691° 

0.66 

*  Zeit.  phys.  chem.,  5,  376  (1890). 
t  Bull.  Soc.  Chim.,  (3),  4,  262  (1890). 


120  THEORETICAL  CHEMISTRY 

Molecular  Volume.  In  dealing  with  the  volume  relations  of 
liquids  it  is  customary  to  employ  the  molecular  volume,  i.e.,  the 
volume  occupied  by  the  molecular  weight  of  the  liquid  in  grams. 
The  justification  of  this  procedure  is  that  when  we  compare  the 
gram-molecular  weights  of  liquids,  the  comparison  involves  equal 
numbers  of  molecules  of  the  different  substances.  Since 


,  mass 

volume  =  j r— ) 

density 


we  may  write 
and  similarly, 


molecular  weight 

molecular  volume  =  -     —3 r—      —  > 

density 


atomic  weight 

atomic  volume  = -= r— 

density 


Relations  between  the  molecular  volumes  of  liquids  were  first 
pointed  out  by  Kopp.*  On  comparing  the  molecular  volumes  of 
different  liquids  at  their  boiling-points,  he  found  that  constant 
differences  in  composition  correspond  to  constant  differences  in 
the  molecular  volumes.  Thus  the  molecular  volumes  of  the 
successive  members  of  an  homologous  series  differ  by  the  same 
number  of  units,  this  difference  corresponding,  for  example,  to  a 
CH2  group.  In  like  manner  the  molecular  volumes  of  various 
groups  have  been  determined,  and  from  these  in  turn  the  atomic 
volumes  of  the  constituent  elements  have  been  worked  out.  The 
atomic  volumes  assigned  by  Kopp  to  some  of  the  elements  com- 
monly entering  into  organic  compounds  are  as  follows :  — 

C  =  11         Cl=22.8  1=37.5       Hydroxyl  O  =  7.8 

#  =  5.5       Br  =  27.8  S  =  22.6       Carbonyl  O  =  12.2 

The  value  of  the  atomic  volume  is  found  to  be  dependent  upon 
the  manner  of  linkage ;  thus  oxygen  in  the  hydroxyl  group  has  the 
atomic  volume,  7.8,  while  oxygen  in  the  doubly  linked  condi- 
tion, as  in  the  carbonyl  group,  has  the  atomic  volume,  12.2.  By 
means  of  such  a  table  of  experimentally  determined  atomic 

*  Lieb.  Ann.,  41,  79  (1842);  96,  153,  303  (1855);  96,  171  (1855). 


LIQUIDS  121 

volumes,  Kopp  showed  that  it  is  possible  to  calculate  the  molec- 
ular volume  of  a  liquid  with  a  fair  degree  of  accuracy.  For 
example,  the  molecular  volume  of  acetic  acid  C2H4O2,  may  be 
calculated  from  the  atomic  volumes  of  its  constituent  atoms  as 
follows :  — 

2  C  =  2  X  11  =      22 

4H  =  4  X  5.5  =    22 
IHydroxylO  =  1  X  7.8  =      7.8 
1  Carbonyl  0  =  1  X  12.2  =  12.2 
Molecular  volume  =  64 . 0 

The  density  of  acetic  acid  at  its  boiling-point  is  0.942,  and  its 
molecular  weight  is  60,  therefore  the  observed  value  of  the  molec- 
ular volume  is  60  -T-  0.942  =  63.7,  a  result  which  is  in  excellent 
agreement  with  that  calculated  from  the  atomic  volumes  of  the 
constituents.  The  more  recent  investigations  of  Thorpe,  Lessen, 
Schiff  and  Buff  afford  a  confirmation  of  the  conclusion  reached  by 
Kopp,  that  the  molecular  volumes  of  liquids  are  in  general  additive. 
While  Kopp  found  that  his  results  were  most  regular  when  the 
molecular  volumes  were  determined  at  the  boiling  temperatures 
of  the  respective  liquids,  the  reason  for  this  did  not  appear  until 
after  van  der  Waals  had  developed  his  theory  of  corresponding 
states.  As  has  been  pointed  out  in  the  preceding  paragraph  the 
boiling-points  of  most  liquids  are  approximately  two-thirds  of 
their  respective  critical  temperatures,  and  therefore  the  boiling- 
points  are  corresponding  temperatures. 

Co-volume.  By  studying  various  series  of  hydrocarbons, 
alcohols  and  ethers,  Traube  *  has  been  led  to  suggest  that  the 
molecular  volume  of  a  liquid  be  looked  upon  as  made  up  of  the 
atomic  volumes  of  its  constituent  elements  and  a  magnitude 
which  he  terms  the  co-volume.  This  latter  he  defines  as  the  space 
surrounding  a  molecule  within  which  it  is  free  to  vibrate  and  from 
which  other  molecules  are  excluded.  The  co-volume  appears  to 
be  nearly  constant  for  a  large  number  of  substances,  its  mean 
value  at  a  temperature  of  15°  C.  being  25.9  cc.  The  values 

*  Uber  den  Raum  der  Atorae.  J.  Traube.  Ahrens'  Sammlung  Chemischer 
und  chemisch-technischer  Vortraege,  4,  255  (1899). 


122  THEORETICAL  CHEMISTRY 

assigned  by  Traube  to  the  atomic  volumes  of  some  of  the  elements 
are  as  follows :  — 

C  =  9.9     0  =    5.5       Br  =  17.7     N  (trivalent)      =    1.5 
H  =  3.1     Cl  =  13.2  1=21.4     N  (pentavalent)  =  10.7 

Traube  has  worked  out  a  series  of  constants  which  must  be  de- 
ducted to  allow  for  ring  formation  and  for  double  and  treble  link- 
ing. By  means  of  these  values,  it  is  possible  to  calculate  the 
molecular  volume  of  a  substance  by  adding  together  the  respective 
atomic  volumes  of  the  constituents  of  the  liquid  and  the  co- 
volume,  25.9.  It  is  of  course  necessary  to  know  the  molecular 
weight  of  the  substance  together  with  its  constitution,  so  that 
due  allowance  may  be  made  for  unsaturation.  For  example, 
the  molecular  volume  of  ethyl  ether,  C^ioO,  may  be  calculated 
by  Traube's  method  as  follows :  — 

4C  =    4X9.9  =  39.6 
10  H  =  10  X  3.1  =  31 
10=     1  X5.5  =    5.5 
76.1 

Co- volume    25 . 9 
Molecular  volume  102.0 

The  molecular  volume,  as  determined  from  the  molecular  weight 
and  density  at  15°  C.,  is  74  -r-  0.7201  =  102.7. 

The  method  of  Traube  may  be  employed  in  roughly  checking 
the  accepted  value  of  the  molecular  weight  of  a  liquid  provided 
its  density  at  15°  C.  is  known,  since  in  the  equation 

M/d  =  2  atomic  volumes  +  25.9,  expressing  Traube's  relation, 
M  is  the  only  unknown  quantity.     It  is  apparent  that  the  liquid 
must  be  non-associated,  since  for  an  associated  substance  the 
normal  co-volume  must  necessarily  accompany  the  polymerized 
molecule.     In  this  case  the  formula  becomes 

M/d  =  2  atomic  volumes  -f  25.9/X 

where  n  denotes  the  number  of  simple  molecules  in  the  polymer. 
Obviously  when  the  molecular  weight  of  a  liquid  is  known,  the 
experimental  determination  of  the  co-volume,  (M/d  —  2  atomic 
vols.)  may  be  used  to  estimate  the  degree  of  association.  The 


LIQUIDS 


123 


values  thus  obtained  are  not  in  satisfactory  agreement  with  the 
factors  of  association  derived  by  means  of  other  methods. 

Refractive  Power  of  Liquids.     The  velocity  of  transmission 
of  light  through  any  medium  depends  upon  its  nature,  especially 


Fig.  27. 

upon  its  density.  When  a  ray  of  light  passes  from  one  medium 
into  another  it  is  refracted,  the  degree  of  refraction  being  such 
that  the  ratio  of  the  sines  of  the  angles  of  incidence  and  refrac- 
tion is  constant  and  characteristic  for  the  two  media.  This 


124 


THEORETICAL  CHEMISTRY 


fundamental  law  of  refraction  was  discovered  by  Snell  about 
1621.  According  to  the  wave  theory  of  light,  the  ratio  of  the 
sines  of  the  angles  of  incidence  and  refraction  is  identical  with 
the  ratio  of  the  velocities  of  light  in  the  two  media.  The  ratio  is 
termed  the  index  of  refraction  and  is  usually  denoted  by  the  letter 
n.  Representing  by  i  and  r,  the  angles  of  incidence  and  refraction, 
and  by  vi  and  v2,  the  respective  velocities  of  light  in  the  two  media, 

we  have 

_  sin  i  _  Vi 
sin  r      v2 

Various  forms  of  apparatus  have  been  devised  for  the  determina- 
tion of  the  refractive  index  of  liquids.  Of  these  the  best  known 
and  most  satisfactory  is  the  refractometer  of  Pulfrich,  an  improved 
form  of  which  is  shown  in  Fig.  27.  While  the  limits  of  this  book 
prohibit  a  detailed  description  of  the  apparatus,  the  fundamental 
principles  involved  in  its  construction  will  be  readily  understood 
from  the  accompanying  diagram,  Fig.  28.  The  liquid  or  fused 
solid  is  placed  in  a  small  glass  cell,  C,  which  is  cemented  to  a  rec- 
tangular prism  of  dense  optical 
glass,  P,  the  refractive  index  of 
which  is  generally  1.61.  A  beam 
of  monochromatic  light,  from  a 
sodium  flame  or  a  spectrum-tube 
containing  hydrogen,  is  allowed  to 
enter  the  prism  in  a  direction  par- 
allel to  the  horizontal  surface  of 
separation  between  the  glass  and 
the  liquid.  After  passing  through 


Fig.  28. 


the  liquid  and  the  prism,  the  beam  emerges  making  an  angle  i  with 
its  original  direction.  By  means  of  a  telescope,  the  emergent  beam 
can  be  observed  and  its  position  noted,  the  angle  of  emergence 
being  read  on  a  divided  circle  attached  to  the  telescope.  From  the 
angle  of  emergence  thus  determined,  the  index  of  refraction  of  the 
liquid  can  be  calculated  in  the  following  manner.  The  value  of 
the  index  of  refraction,  N,  for  air/glass  being  known,  we  have 


N  = 


smi 
sinr 


(1) 


LIQUIDS  125 

The  angle  of  incidence  of  the  last  ray  entering  the  prism  from  the 
liquid  is  90°,  or  sin  ii  =  1.  The  index  of  refraction,  n\y  for  liquid/ 
glass  may  be  calculated  thus, 


sm  ri      sin  7*1 
But 


sn  7*1  =  cos  r 


=  V  1  —  sin2  r.  (3) 

Transposing  equation  (1)  and  substituting  in  equation  (3),  we 
have 


or 


sin  ri***  VN2-sin?i.  (4) 


Therefore,  substituting  equation  (4)  in  equation  (2),  we  have 

N 


sn 


Remembering  that  n  =  N/rii,  we  have  for  the  index  of  refraction, 
n,  for  air/liquid,  by  substitution  in  equation  (5), 


n  =  v       —  sin2z, 


or  if  N  =  1.61,  _ 

n  =  V2.5921  -sin2!. 


The  values  of  VAT2  —  sin2  i  are  generally  given,  for  different  values 
of  i,  in  tables  supplied  with  the  refractometer,  thus  saving  the 
experimenter  a  somewhat  laborious  calculation.  The  value  of  n 
thus  obtained  is  the  index  of  refraction  from  air  into  the  liquid;  if 
the  index  from  vacuum  into  the  liquid,  the  so-called  absolute  index, 
is  required,  the  value  of  n  must  be  multiplied  by  1.00029.  The 
index  of  refraction  is  dependent  upon  temperature,  pressure  and 
in  general  upon  all  conditions  which  affect  the  density  of  the 
medium.  Furthermore,  it  is  dependent  upon  the  wave-length  of 
the  light  employed,  the  index  for  the  red  rays  being  greater  than 
that  for  the  violet  rays.  It  is  therefore  necessary  in  making 


126  THEORETICAL  CHEMISTRY 

measurements  of  refractive  indices  to  use  light  of  a  definite  wave- 
length, or  what  is  termed  monochromatic  light.  The  sodium 
flame  is  most  frequently  used  for  this  purpose,  the  wave-length 
being  represented  by  the  letter  D.  Measurements  of  the  refrac- 
tive index  referred  to  the  D-line  of  sodium  are  commonly  desig- 
nated by  the  symbol  UD-  When  incandescent  hydrogen  is  employed 
as  a  source  of  light,  the  refractive  index  may  be  determined 
for  the  C-,  F-  and  G-lines,  the  respective  values  being  represented 
by  nc,  np,  and  no- 

Specific  and  Molecular  Refraction.  Various  attempts  have 
been  made  to  express  the  refractive  power  of  a  liquid  by  a  formula 
which  is  independent  of  variations  of  temperature  and  pressure. 
Of  the  different  formulas  proposed  but  two  need  be  mentioned. 
The  first,  due  to  Gladstone  and  Dale,*  is  as  follows:  — 

n-  1 


in  which  d  denotes  the  density  of  the  liquid  and  rx  is  the  so-called 
specific  refraction.  The  other  formula,  proposed  by  Lorenz  f  and 
Lorentz,J  has  the  following  form:  — 


= 


This  formula  is  superior  to  that  of  Gladstone  and  Dale  which  is 
purely  empirical.  It  is  based  upon  the  electromagnetic  theory 
of  light  and  gives  values  of  r2  which  are  quite  independent  of  the 
temperature.  In  order  that  we  may  compare  the  refractive 
powers  of  different  liquids,  the  specific  refractions  are  multiplied 
by  their  respective  molecular  weights,  the  resulting  products 
being  termed  their  molecular  refractions.  As  the  result  of  a  large 
number  of  experiments,  it  has  been  shown  that  the  molecular 
refraction  of  a  compound  is  made  up  of  the  sum  of  the  refractive 
constants  of  the  constituent  atoms,  or  in  other  words  refractive 
power  is  an  additive  property.  The  values  of  the  refractive  con- 
stants of  the  elements  and  commonly-occurring  groups  have  been 

*  Phil.  Trans.  (1858). 

t  Wied.  Ann.,  n,  70  (1880). 

t  Ibid.,  9,  641  (1880). 


LIQUIDS  127 

determined  with  great  care  by  Briihl  and  others,  the  method 
employed  being  similar  to  that  used  by  Kopp  in  connection  with 
his  investigations  on  molecular  volumes.  Thus,  Briihl  found  in 
the  homologous  series  of  aliphatic  compounds  that  a  difference 
of  CH2  in  composition  corresponds  to  a  constant  difference  of 
4.57  in  molecular  refraction.  Then,  having  determined  the 
molecular  refraction  of  a  ketone  or  an  aldehyde  of  the  composition, 
CnH2nO,  he  subtracted  n  times  the  value  of  CH2  and  obtained  the 
atomic  refraction  of  carbonyl  oxygen.  By  deducting  the  molecular 
refraction  of  the  hydrocarbon,  CnH2n+2,  from  that  of  the  corre- 
sponding alcohol,  CnHtn+zO,  he  obtained  the  atomic  refraction  of 
hydroxyl  oxygen.  By  subtracting  six  times  the  value  of  CH2 
from  the  molecular  refraction  of  hexane,  C6Hi4,  he  obtained  tne 
refractive  constant  for  hydrogen  or  2  H  =  2.08.  In  like  manner 
the  refractions  of  other  elements  and  groups  of  elements  were 
determined. 

Just  as  in  the  case  of  molecular  volumes  so  with  molecular 
refractions,  the  arrangement  of  the  atoms  in  the  molecule  must 
be  taken  into  consideration.  Bruhl,*  who  has  devoted  much  time 
to  the  investigation  of  the  effect  of  constitution  upon  refraction, 
has  pointed  out  that  the  molecular  refraction  of  compounds  con- 
taining double  and  triple  bonds  is  greater  than  the  calculated  value, 
and  he  has  assigned  to  these  bonds  definite  constants  of  refrac- 
tion. The  values  of  the  atomic  refractions  for  a  few  of  the  elements 
as  given  by  Briihl  are  as  follows :  — 

C=    2.48  Hydroxyl  0    =  1.58 

H=    1.04  Carbonyl  O    =2.34 

Cl  =    6.02  Double  bond  =  1.78 

I  =  13.99  Triple  bond    =  2.18 

More  recent  investigations  bring  out  the  fact  that,  when  double 
or  triple  bonds  occupy  adjacent  positions  in  the  molecule,  the 
simple  additive  relations  no  longer  obtain.  The  determination 
of  the  molecular  refraction  of  a  liquid  affords  a  means  of  ascertain- 
ing or  confirming  its  chemical  constitution.  For  example,  geraniol 
has  the  formula  CioHi80,  and  its  chemical  behavior  is  such  as  to 

*  Proc.  Roy.  Inst.,  18,  122  (1906). 


128  THEORETICAL  CHEMISTRY 

warrant  the  conclusion  that  it  is  a  primary  alcohol.  The  value 
of  UD  is  1.4745,  from  which  the  molecular  refraction  is  calcu- 
lated to  be  48.71.  The  molecular  refraction  calculated  from  the 
atomic  refractions  given  in  the  preceding  table  is:  - 

IOC  =  10X2.48  =  24.80 

18H  =  18  X  1.04  =  18.72 

1  Hydroxyl  0  =    1  X  1.58  =    1.58 

Molecular  refraction  45.  10 

The  difference  between  the  theoretical  and  experimental  values 
of  the  molecular  refraction  is  48.71  —  45.10  =  3.61,  which  is 
approximately  twice  the  value  of  a  double  bond,  1.78  X  2  =  3.56. 
From  this  we  conclude  that  the  molecule  of  geraniol  contains  two 
double  bonds.  Furthermore  an  alcohol  of  the  formula,  Ci0Hi8O, 
containing  two  double  bonds  cannot  possess  a  ring  structure  and 
therefore  must  be  a  member  of  the  aliphatic  group  of  compounds. 
This  conclusion  is  supported  by  the  chemical  properties  of  the 
substance.*  In  a  similar  manner  the  Kekul6  formula  for  benzene 
has  been  confirmed,  the  difference  between  the  theoretical  and 
experimental  values  of  the  molecular  refraction  indicating  the 
presence  of  three  double  bonds  in  the  molecule. 

Specific  Refraction  of  Mixtures.  The  specific  refraction  of  an 
homogeneous  mixture  or  solution  is  the  mean  of  the  specific 
refractions  of  its  constituents.  Thus,  if  the  specific  refractions 
of  the  mixture  and  its  two  components  are  represented  by  ri,  r2, 
and  r3,  then 

p        ,    (100  -p) 

riS=I66rH    ~~~ 


where  p  denotes  the  percentage  of  the  constituent  whose  specific 
refraction  is  r2.  Hence  it  is  possible  to  determine  the  specific 
refraction  of  a  substance  in  solution  by  measuring  the  refractive 
indices  and  densities  of  the  solution  and  solvent.  If  the  refrac- 
tive indices  of  the  solvent,  solution  and  dissolved  substance  are 
*  The  accepted  structural  formula  of  geraniol  is 

H  -  C  -  CH2OH 
(CH3)2C  =  CH-CH2.CH2  -  C  -  CH8 


LIQUIDS  129 

represented  by  wi,  HZ,  and  ns  respectively,  and  if  d\t  ck,  and  ds 
denote  the  corresponding  densities,  then  we  have 

g2  -  1      100  -  p  nf  -  1 


where  p  is  the  percentage  of  the  dissolved  substance.  As  has 
already  been  mentioned,  the  formula  of  Lorenz-Lorentz  is  based 
upon  the  electromagnetic  theory  of  light.  According  to  this 
theory  n2  —  l/n2  +  2  expresses  the  fraction  of  the  unit  of  volume 
of  the  substance  which  is  actually  occupied  by  it.  From  this  it 

M  n2  —  1 

follows  that  the  molecular  refraction,  -r  •  -^—  —  r,  is  an  expression 

a    n^  +  2. 

of  the  volume  actually  occupied  by  the  atomic  nuclei  of  the 
molecule.  It  is  interesting  to  note  that  the  ratio  of  the  sum  of 
the  atomic  volumes,  calculated  by  the  method  of  Traube,  to  the 
corrected  molecular  volume,  as  determined  by  the  Lorenz-Lorentz 
formula,  is  approximately  constant,  or 

£  atomic  volumes 

-  TT  —  2  _  -  =3.45  approximately. 


This  may  be  considered  as  the  ratio  of  the  volume  within  which 
the  atoms  execute  their  vibrations  to  their  actual  material  volume. 
Rotation  of  the  Plane  of  Polarized  Light.  Some  liquids  when 
placed  in  the  path  of  a  beam  of  polarized  light  possess  the  prop- 
erty of  rotating  the  .plane  of  polarization  to  the  right  or  to  the 
left.  Such  liquids  are  said  to  be  optically  active.  Those  substances 
which  rotate  the  plane  of  polarization  to  the  right  are  termed 
dextro-rotatory,  while  those  which  cause  an  opposite  rotation  are 
called  levo-rotatory.  The  determination  of  the  rotatory  power  of 
a  liquid  is  made  by  means  of  an  instrument  known  as  a  polarimeter, 
a  convenient  form  of  which  is  shown  in  Fig.  29.  The  essential 
parts  of  this  instrument  are  two  similar  Nicol  prisms  placed  one 
behind  the  other  with  their  axes  in  the  same  straight  line.  The 
light  after  passing  through  the  forward  prism,  P,  known  as  the 
polarizer,  has  its  vibrations  reduced  to  a  single  plane;  it  is  said 
to  be  plane  polarized.  On  entering  the  rear  Nicol  prism,  A, 


130 


THEORETICAL  CHEMISTRY 


known  as  the  analyzer,  the  light  will  either  pass  through  or  be 
completely  stopped,  depending  upon  the  position  of  the  prism. 
If  the  analyzer  be  slowly  rotated,  it  will  be  observed  that 
the  positions  of  maximum  transmission  and  extinction  occur  at 
points  90°  apart.  If  the  analyzer  be  rotated,  so  that  its  axis  is 
at  right  angles  to  the  axis  of  the  polarizer,  the  field  observed  will 
be  dark,  no  light  being  transmitted.  If  now  a  tube  similar  to 
that  shown  in  Fig.  30  be  filled  with  an  optically  active  liquid 
and  placed  between  the  polarizer  and  analyzer,  the  field  will 


Fig.  29. 

become  light  again,  due  to  the  rotation  of  the  plane  of  polariza- 
tion by  the  optically-active  substance.  The  extent  to  which  the 
plane  of  polarization  has  been  rotated  can  be  determined  by 
turning  the  analyzer  until  the  field  becomes  dark  again,  and  read- 
ing on  the  divided  circle,  K,  the  number  of  degrees  through  which 
it  has  been  moved.  When  it  is  necessary  to  turn  the  analyzer 
to  the  right,  the  substance  is  dextro-rotatory,  and  when  it  is  neces- 
sary to  turn  it  to  the  left,  the  substance  is  levo-rotatory.  Various 


LIQUIDS  131 

optical  accessories  have  been  added  to  the  simple  polarimeter 
described  above  to  render  the  instrument  more  sensitive,  but  for 
these  details  the  student  must  consult  some  special  treatise.* 
The  angle  of  rotation  is  dependent  upon  the  nature  of  the  liquid, 
the  length  of  the  column  of  substance  through  which  the  light 
passes,  the  wave-length  of  the  light  used,  and  the  temperature  at 
which  the  measurement  is  made.  It  is  customary  in  polarimetric 


Fig.  30. 

work  to  employ  sodium  light  and,  unless  otherwise  specified,  it 
may  be  assumed  that  a  given  rotation  corresponds  to  the  D-line. 

Specific  and  Molecular  Rotation.  The  results  of  polarimetric 
measurements  are  expressed  either  as  specific  rotations  or  as 
molecular  rotations,  the  latter  being  preferable  since  the  optical 
activities  of  different  substances  may  then  be  compared. 

The  specific  rotation  is  obtained  by  dividing  the  observed  rota- 
tion by  the  product  of  the  length  of  the  column  of  liquid  and  its 
density,  or 

r   i  a 

Wi  =  w' 

where  [a]t  is  the  specific  rotation  at  the  temperature,  t,  a  the 
observed  angle,  I,  the  length  of  the  column  of  liquid  in  decimeters 
and  d,  its  density.  If  the  specific  rotation  is  multiplied  by  the 
molecular  weight  of  the  substance,  the  molecular  rotation  is  ob- 
tained, but  owing  to  the  fact  that  the  resulting  numbers  are  too 
large,  it  is  customary  to  express  the  molecular  rotation  as  one 
one-hundredth  of  this  value,  thus 

Ma 


The  specific  and  molecular  rotations  of  solutions  of  optically 
active  substances  may  also  be  determined,  if  we  assume  that  the 

*  See  for  example,  "  The  Optical  Rotatory  Power  of  Organic  Substances 
and  its  Practical  Applications."     H.  Landolt,  trans,  by  J.  H.  Long. 


132 


THEORETICAL  CHEMISTRY 


solvent  is  without  effect.  While  this  assumption  is  justifiable 
with  aqueous  solutions,  it  is  not  so  when  non-aqueous  solvents  are 
used.  If  g  grams  of  an  optically  active  substance  be  dissolved  in 
v  cc.  of  solvent,  then 

av  M    av 

[*!«-•£,     and     [«M]<  =  JOO-^> 

or  if  the  composition  of  the  solution  is  expressed  in  terms  of 
weight  instead  of  volume,  g  grams  of  substance  being  dissolved 
in  100  grams  of  solution  of  density  d}  then 


100  « 

gdl 


and     [aM]t  = 


Ma 

gdl 


Optical  Activity  and  Chemical  Constitution.  The  fact  that 
some  substances  have  the  power  of  rotating  the  plane  of  polarized 
light  was  first  discovered  by  Biot,  but  the  credit  for  recognizing 
the  chemical  significance  of  this  fact  belongs  to  Pasteur.*  He 
discovered  that  ordinary  recemic  acid  can  be  separated  into  two 
optically  active  modifications,  one  of  which  is  dextro-  and  the 
other  levo-rotatory,  the  numerical  values  of  the  two  rotations 


Fig.  31. 

being  identical.  If  a  solution  of  sodium  ammonium  racemate 
be  allowed  to  evaporate  at  a  low  temperature,  crystals  of  the 
composition  NaNH^EUOe  •  4  H2O  will  separate.  On  close  in- 
spection it  will  be  found  that  the  crystals  are  not  all  alike,  but 
that  they  may  be  divided  into  two  classes,  one  class  showing 
some  unsymmetrical  crystal  surfaces  which  are  oppositely  placed 
in  the  crystals  of  the  other  class.  The  crystals  of  one  class  may 

*  Ann.  Chim.  Phys.  (3),  24,  442  (1848);  28,  56  (1850);  31,  67  (1851). 


LIQUIDS  133 

be  regarded  as  the  mirror  images  of  those  of  the  other  class: 
such  crystals  are  said  to  be  enantiomorphous.  The  forms  usually 
assumed  by  the  two  enantiomorphous  modifications  of  sodium 
ammonium  reacemate  are  shown  in  Fig.  31.  After  separating 
the  two  forms  Pasteur  dissolved  each  in  water,  making  the  solu- 
tions of  the  same  strength.  The  solution  of  the  crystals  with  the 
" right-handed  faces"  was  found  to  be  dextro-rotatory,  while  that 
of  the  crystals  with  the  " left-handed  faces"  was  found  to  be  levo- 
rotatory.  Pasteur  then  decomposed  the  two  salts  obtained  from 
sodium  ammonium  racemate  and  obtained  the  corresponding 
acids,  which  he  called  dextro-  and  levo-racemic  acids.  It  was 
subsequently  shown  that  the  two  acids  were  identical  with  dextro- 
and  levo-tartaric  acids.  Finally,  when  Pasteur  mixed  equiv- 
alent amounts  of  concentrated  solutions  of  dextro-and  levo- 
tartaric  acids,  an  appreciable  evolution  of  heat  was  observed, 
indicating  that  a  chemical  reaction  had  taken  place.  After 
allowing  the  solution  to  stand  for  some  time,  crystals  of  ordinary 
racemic  acid  were  obtained.  Thus  it  wus  clearly  proven  that  an 
optically  inactive  substance  may  be  separated  into  two  opti- 
cally active  modifications,  possessing  equal  and  opposite  rotatory 
powers,  and  that  by  mixing  equivalent  quantities  of  the  two 
optically  active  forms,  the  optically  inactive  substance  may  be 
recovered. 

Pasteur  discovered  and  applied  three  other  methods  in  addi- 
tion to  the  mechanical  method  already  described,  for  the  separation 
of  a  substance  into  its  optically  active  modifications.  These  are 
as  follows:  —  (a)  Method  of  Crystallization;  (b)  Method  of  Forma- 
tion of  Derivatives ;  and  (c)  Method  of  Ferments. 

Method  of  Crystallization.  To  a  supersaturated  solution  of  the 
racemic  modification  a  very  small  crystal  of  one  of  the  active 
forms  is  added.  This  will  induce  the  separation  of  crystals  of 
the  same  form,  inoculation  with  a  dextro-crystal  producing  the 
dextro-form  and  inoculation  with  a  levo-crystal  producing  the 
levo-form. 

Method  of  Formation  of  Derivatives.  In  this  method  an  optic- 
ally active  substance,  generally  an  alkaloid,  is  added  to  the  racemic 
modification,  producing  optically  active  derivatives  having  differ- 


134  THEORETICAL  CHEMISTRY 

ent  solubilities.  Thus  if  cinchonine,  an  optically  active  alkaloid 
having  the  formula,  CuK^O,  be  added  to  the  racemic  modifica- 
tion of  tartaric  acid,  the  cinchonine  salt  of  the  levo-acid  will 
crystallize  first.  The  crystals  of  the  cinchonine  salt  are  then 
removed  and  after  adding  ammonia  to  displace  the  alkaloid, 
dilute  sulphuric  acid  is  added  and  the  pure  levo-tartaric  acid  is 
obtained. 

Method  of  Ferments.  Notwithstanding  the  fact  that  optical 
antipodes  resemble  each  other  so  closely  in  most  of  their  properties, 
Pasteur  found  that  certain  micro-organisms  have  the  power  of 
distinguishing  sharply  between  these  forms.  For  example,  if 
penicillium  glaucum  be  introduced  into  a  solution  of  racemic 
tartaric  acid,  it  thrives  at  the  expense  of  the  dextro-acid  and 
eventually  leaves  the  pure  levo-form.  In  this  method  one  of  the 
active  modifications  is  always  lost. 

Pasteur  was  the  first  to  point  out  that  there  must  be  some  inti- 
mate connection  between  optical  activity  and  the  constitution  of 
the  molecule.  It  remained  for  Le  Bel  *  and  van't  Hoff  f  to  for- 
mulate independently  and  almost  simultaneously  an  hypothesis 
to  account  for  optical  activity  on  the  basis  of  molecular  constitu- 
tion. Their  important  work  laid  the  foundation  of  spatial  chemis- 
try, commonly  termed  stereochemistry  (derived  from  the  Greek 
0-re/oeos  =  a  solid).  Le  Bel  accepted  Pasteur's  view  that  optical 
activity  is  dependent  upon  a  condition  of  asymmetry,  but  whether 
this  asymmetry  is  a  property  of  the  crystal  alone  or  whether  it 
belongs  to  the  molecule  of  the  optically  active  substance,  was  the 
question  he  set  himself  to  answer.  He  found,  on  dissolving  certain 
optically  active  crystals  in  an  inactive  solvent,  that  the  optical 
activity  is  imparted  to  the  solution  and  therefore  he  concluded 
that  the  condition  of  asymmetry  must  exist  in  the  chemical  mole- 
cule. All  of  the  optically  active  substances  known  to  Le  Bel  were 
compounds  of  carbon.  An  examination  of  the  formulas  of  these 
compounds  led  him  to  ascribe  the  cause  of  their  optical  activity 
to  the  presence  of  an  asymmetric  carbon  atom,  that  is,  a  carbon 
atom  combined  with  four  different  atoms  or  groups  of  atoms. 

*  Bull.  Soc.  Chim.  (2),  22,  337  (1874). 

t  Ibid.  (2),  23,  295  (1875). 


LIQUIDS 


135 


One  of  the  simplest  examples  is  afforded  by  lactic  acid,  the  struc- 
tural formula  of  which  is 

H 
I 

CH3  -  C  -  COOH 
I 
OH 

In  this  formula  the  asymmetric  carbon  atom  is  placed  at  the 
center  and  is  in  combination  with  hydrogen,  hydroxyl,  methyl 
and  carboxyl.  In  connection  with  his  work  on  the  relation 
between  optical  activity  and  asymmetry,  Le  Bel  pointed  out  that 
active  forms  never  result  from  laboratory  syntheses,  the  racemic 
modification  being  invariably  obtained.  Van't  Hoff  reached  con- 
clusions similar  to  those  of  Le  Bel  and  proposed  the  additional 


COOH 


COOH 


Fig.  32. 


theory  of  the  asymmetric  tetrahedral  carbon  atom.  Since  the  four 
valences  of  the  carbon  atom  are  equivalent,  as  the  work  of  Henry 
on  methane  has  shown  them  to  be,  van't  Hoff  pointed  out  that 
the  only  possible  geometrical  arrangement  of  the  atoms  in  the 
molecule  of  methane,  must  be  that  in  which  the  carbon  atom  is 
placed  at  the  center  of  a  regular  tetrahedron  with  the  four 
hydrogen  atoms  at  the  four  apices.  He  then  pointed  out  that 
when  the  four  valences  of  the  tetrahedral  carbon  atom  are  satis- 
fied with  different  atoms  or  groups,  no  plane  of  symmetry  can  be 
passed  through  the  figure,  the  carbon  atom  being  asymmetric. 
This  conception  of  Le  Bel  and  van't  Hoff  forms  the  basis  of  all 
stereochemistry,  and  has  proved  of  inestimable  value  to  the 
organic  chemist  in  enabling  him  to  explain  the  existence  of  many 
isomeric  compounds.  Thus,  ordinary  lactic  acid  can  be  split  into 


136 


THEORETICAL  CHEMISTRY 


two  optically  active  isomers.  Aside  from  the  fact  that  one  acid 
is  dextro-  and  the  other  is  levo-rotatory,  the  properties  of  the 
two  acids  are  practically  identical.  If  the  formulas  are  written 
spatially,  the  different  groups  can  be  arranged  about  the 
asymmetric  carbon  atom  in  such  a  way  that  the  two  tetrahedra 
shall  be  mirror  images  of  each  other,  as  shown  in  Fig.  32.  It  will 
be  observed  that  these  two  tetrahedra  can  in  no  way  be  super- 
posed so  that  the  same  groups  fall  over  each  other,  that  is  to 
say,  they  are  enantiomorphous  forms.  In  tartaric  acid  there  are 
two  asymmetric  carbon  atoms  as  is  evident  when  its  structural 
formula  is  written  as  follows:  — 

H      H 
I        I 

HOOC  -  C  -  C  -  COOH 
I        I 
OH   OH 

If  the  stereochemical  formulas  of  the  dextro-  and  levo-acids  be 
represented  as  in  Fig.  33,  (a)  and  (b),  it  will  be  apparent  that 
the  theory  admits  of  the  existence  of  another  isomer  with  the 
atoms  and  groups  arranged  as  in  Fig.  33  (c). 

Racemic  Acid 
A 


COOH 
d-Tartarte  Ae® 

«*) 


COOH 
Z-Tartaric  Acid 

ib) 
Fig.  33. 


OH 


COOH 

Meao-Tartaric  Acid 

(C) 


In  this  arrangement  the  asymmetry  of  the  upper  tetrahedron 
is  the  reverse  of  that  of  the  lower,  and  consequently  the  optical 
activity  of  one-half  of  the  molecule  exactly  "compensates  the  optical 


LIQUIDS  137 

activity  of  the  other  half,  and  the  molecule  as  a  whole  is  inactive. 
It  is  evident  that  such  a  tartaric  acid  could  not  be  split  into  two 
active  forms.  Actually  there  are  four  tartaric  acids  known,  viz., 
(1)  inactive  racemic  acid  which  is  separable  into  (2)  dextro-tartaric 
acid  and  (3)  levo-tartaric  acid;  and  (4)  meso-tartaric  acid,  an  in- 
active substance  which  has  never  been  separated  into  two  active 
forms,  but  which  has  the  same  formula,  the  same,  molecular 
weight  and  in  general  the  same  properties  as  the  dextro-  or  levo- 
tartaric  acids.  Inactive  forms,  such  as  meso-tartaric  acid,  are  said 
to  be  inactive  by  internal  compensation.  This  constitutes  one  of 
many  beautiful  confirmations  of  the  van't  Hoff  theory  of  the 
asymmetric  tetrahedral  carbon  atom. 

Meso-tartaric  acid  furnishes  an  illustration  of  the  fact  that 
asymmetric  carbon  atoms  may  be  present  in  the  molecule  with- 
out imparting  optical  activity  to  the  substance.  The  converse 
of  this  proposition,  however,  that  optical  activity  is  dependent 
upon  asymmetric  carbon  atoms,  is  generally  true.  Quite  recently 
some  substances  apparently  containing  no  asymmetric  carbon 
atoms  have  been  discovered  which  are  optically  active.  An 
example  of  such  a  substance  is  1-methyl  cyclohexylidene-4  acetic 
acid,  t&  which  the  following  formula  has  been  assigned:  — 


CH3CH  )C:CH-COOH 


Other  atoms  aside  from  carbon  may  be  asymmetric;  thus  certain 
compounds  of  nitrogen,  sulphur  and  tin  have  been  shown  to  be 
optically  active.  The  theory  also  furnishes  an  explanation  of 
the  fact,  pointed  out  by  Le  Bel,  that  optically  active  forms  are 
never  obtained  by  direct  synthesis.  Since  the  rotatory  power  is 
dependent  upon  the  arrangement  of  the  atoms  and  groups  in  the 
molecule,  it  follows  from  the  doctrine  of  probability  that  as  many 
dextro  as  levo  configurations  will  be  formed  and  consequently  the 
racemic  modification  will  be  obtained.  Up  to  the  present  time  no 
satisfactory  generalization  has  been  discovered  as  to  the  factors 
determining  the  molecular  rotation  in  any  particular  case.  An 


138  THEORETICAL  CHEMISTRY 

attempt  in  this  direction  has  been  made  by  Guye,*  in  which  he 
ascribes  the  magnitude  of  the  observed  rotation  to  the  relative 
masses  of  the  atoms  or  groups  which  are  in  combination  with  the 
tetrahedral  carbon  atom.  But  it  cannot  be  mass  alone  which 
conditions  optical  activity,  since  substances  are  known  which 
rotate  the  plane  of  polarization  notwithstanding  the  fact  that 
their  molecules  have  two  groups  of  equal  mass  in  combination 
with  the  asymmetric  carbon  atom.  The  molecular  rotations  of 
the  members  of  homologous  series  exhibit  some  regularities,  but 
on  the  other  hand  many  exceptions  occur  which  cannot  be  satis- 
factorily explained.  About  all  that  can  be  said  at  the  present 
time  is,  that  optical  activity  is  a  constitutive  property. 

Magnetic  Rotation.  That  many  substances  acquire  the  power 
of  rotating  the  plane  of  polarized  light  when  placed  in  an  intense 
magnetic  field  was  first  observed  by  Faraday  f  in  1846. 

The  relation  between  chemical  composition  and  magnetic 
rotatory  power  has  since  been  investigated  very  exhaustively  by 
W.  H.  Per  kin,  |  his  experiments  in  this  field  having  been  continued 
for  more  than  fifteen  years.  In  brief,  Perkin's  method  of  investi- 
gating magnetic  rotatory  power  consisted  in  introducing  the 
liquid  to  be  examined  into  a  polarimeter  tube  1  decimeter  in 
length  and  then  placing  the  tube  axially 
betweeen  the  perforated  poles  of  a 
powerful  electromagnet,  as  shown  in 
Fig.  34.  Upon  exciting  the  magnet  it 
was  found  that  the  plane  of  polarization 
had  been  rotated,  either  to  the  right 
or  the  left,  the  direction  of  rotation 
depending  upon  the  direction  of  the 
current,  the  intensity  of  the  magnetic 

field  and  the  nature  of  the  liquid.  Perkin  used  the  sodium  flame 
as  his  source  of  light  and  carried  out  all  of  his  experiments 
at  15°  C.  He  expressed  his  results  by  means  of  the  formula, 

*  Compt.  rend.,  no,  714  (1890). 

t  Phil.  Trans.,  136,  1  (1846). 

t  Jour,  prakt.  Chem.  [2],  31,  481  (1885);  Jour.  Chem.  Soc.,  49,  777;  41, 
808;  53,  561,  695;  59,  981;  61,  287,  800;  63,  57;  65,  402,  815;  67,  255;  60, 
1025  (1886-1896). 


LIQUIDS  139 

Ma/d,  a  being  the  observed  angle  of  rotation,  d,  the  density 
of  the  liquid  and  M  its  molecular  weight.  All  measurements 
were  expressed  in  terms  of  water  as  a  standard:  thus  if  Ma/d 
is  the  rotation  for  any  substance  and  M'a'/d'  is  the  corre- 
sponding rotation  for  water,  then,  according  to  Perkin,  the 
molecular  magnetic  rotation' will  be  given  by  the  ratio,  Ma/d  : 
M'a'/d'  or  Mad'/M'a'd. 

The  molecular  magnetic  rotation  for  a  large  number  of  organic 
compounds  has  been  determined  by  Perkin,  who  has  shown  it  to 
be  an  additive  property.  In  any  one  homologous  series  the  value 
of  the  molecular  magnetic  rotation  is  given  by  the  formula 

mol.  mag.  rotation  =  a  +  rib, 

where  a  is  a  constant  characteristic  of  the  series,  b  is  a  constant 
corresponding  to  a  difference  of  CH2  in  composition,  its  value 
being  1.023,  and  n  is  the  number  of  carbon  atoms  contained  in 
the  molecule.  This  formula  is  applicable  only  to  compounds 
which  are  strictly  homologous,  isomeric  substances  in  two  differ- 
ent series  having  quite  different  rotations.  The  constitution  of 
the  molecule  exerts  as  great  an  influence  on  magnetic  rotation 
as  it  does  on  refraction,  a  double  bond  causing  an  appreciable 
increase  in  the  value  of  a.  The  results  of  experiments  on  mag- 
netic rotation  show  that  nothing  like  the  same  regularities  exist 
as  have  been  discovered  for  molecular  refraction  and  molecular 
volume.  The  rotatory  powers  of  various  inorganic  substances 
have  been  determined,  but  the  results  are  too  irregular  to  admit 
of  any  satisfactory  interpretation. 

Absorption  Spectra.  When  a  beam  of  white  light  is  passed 
through  a  colored  liquid  or  solution  and  the  emergent  beam  is 
examined  with  a  spectroscope,  a  continuous  spectrum  crossed  by 
a  number  of  dark  bands  is  obtained.  A  portion  of  the  light  has 
been  absorbed  by  the  liquid.  Such  a  spectrum  is  known  as  an 
absorption  spectrum.  If  instead  of  passing  the  light  through  a 
liquid  it  is  passed  through  an  incandescent  gas,  a  spectrum  will 
be  obtained  which  is  crossed  by  numerous  fine  lines,  termed 
Fraunhofer  lines.  Such  lines  occupy  the  same  positions  as  the 
corresponding  colored  lines  in  the  emission  spectrum  of  the  gas. 


140 


THEORETICAL  CHEMISTRY 


It  follows,  therefore,  that  the  absorption  spectrum  is  quite  as  char- 
acteristic of  a  substance  as  its  emission  spectrum,  and  from  a 
careful  study  of  the  absorption  spectra  of  liquids  we  may  expect 
to  gain  some  insight  into  their  molecular  constitution.  The 
pioneer  workers  in  this  field  were  Hartley  and  Baly  *  and  it  is 
largely  to  them  that  we  owe  our  present  experimental  methods. 
The  instrument  employed  for  photographing  spectra  is  called  a 


Fig.  35. 

spedrograph,  a  very  satisfactory  form  being  shown  in  Fig.  35. 
It  differs  from  an  ordinary  spectroscope  in  that  the  eye-piece  is 
replaced  by  a  photographic  camera.  This  attachment  is  clearly 
shown  in  the  illustration.  The  plateholder  is  so  constructed  that 
only  a  narrow  horizontal  strip  of  the  plate  is  exposed  at  any  one 
time,  thus  making  it  possible  to  take  a  series  of  photographs  on 
the  same  plate  by  simply  lowering  the  holder.  By  means  of  a 
millimeter  scale,  also  shown  in  the  illustration,  the  plateholder 
can  be  moved  through  the  same  distance  each  time  before  expos- 
*  See  numerous  papers  in  the  Jour.  Chem.  Soc.,  since  1880. 


•  LIQUIDS  141 

ing  a  fresh  portion  of  the  plate,  thus. insuring  an  equally-spaced 
series  of  spectrum  photographs.  In  order  that  spectra  in  the  ultra- 
violet region  may  be  photographed,  it  is  customary  to  equip  the  in- 
strument with  quartz  lenses  and  a  quartz  prism,  ordinary  glass  not 
being  transparent  to  the  ultra-violet  rays.  Using  a  spectrograph 
furnished  with  a  quartz  optical  system,  it  is  possible  to  photograph 
on  a  single  plate  the  entire  spectrum  from  2000  to  8000  Angstrom 
units.  A  scale  of  wave-lengths  photographed  on  glass  is  provided 
with  the  instrument  so  that  the  wave-lengths  of  lines  or  bands 
can  be  read  off  directly  by  laying  the  scale  over  the  photographs. 

The  source  of  light  to  be  used  depends  upon  the  character  of 
the  investigation.  If  a  source  rich  in  ultra-violet  rays  is  desired, 
the  light  from  the  electric  spark  obtained  between  electrodes  pre- 
pared from  an  alloy  of  cadmium,  lead  and  tin  is  very  satisfactory; 
or  the  light  from  an  arc  burning  between  iron  electrodes  may  be 
used.  For  investigations  in  the  visible  region  of  the  spectrum 
the  Nernst  lamp  is  unsurpassed.  In  using  the  spectrograph  for 
the  purpose  of  studying  the  constitution  of  a  dissolved  substance, 
it  is  necessary  to  determine  not  only  the  number  and  position  of 
the  absorption  bands,  but  also  the  persistence  of  these  bands  as 
the  solution  is  diluted. 

According  to  Beer's  law  the  product  of  the  thickness,  t,  of  an 
absorbing  layer  of  solution  of  molecular  concentration,  ra,  is  con- 
stant, or  mt  =  k.  If  then  the  thickness  of  a  given  layer  of  solu- 
tion is  diminished  n  times,  its  absorption  will  be  the  same  as  that 
of  a  solution  whose  concentration  is  only  I/ nth  of  that  of  the  original 
solution.  Thus,  by  varying  the  thickness  of  the  absorbing  layer 
we  can  produce  the  same  effect  as  by  changing  the  concentration. 
The  convenient  device  of  Baly  for  altering  the  length  of  the 
absorbing  column  of  liquid  is  shown  in  Fig.  36  attached  to  the 
collimator  of  the  spectrograph.  It  consists  of  two  closely-fitting 
tubes,  one  end  of  each  tube  being  closed  by  a  plane,  quartz  disc. 
The  outer  tube  is  fitted  with  a  small  bulbed-f  unnel  and  is  graduated 
in  millimeters.  The  two  tubes  are  joined  by  means  of  a  piece  of 
rubber  tubing  which  prevents  leakage  of  the  contents,  and  at  the 
same  time  admits  of  the  adjustment  of  the  column  of  liquid  to 
the  desired  length  by  simply  sliding  the  smaller  tube  in  or  out. 


142 


THEORETICAL  CHEMISTRY 


Molecular  Vibration  and  Chemical  Constitution.  There  are 
two  systems  of  graphic  representation  of  the  results  of  spectro- 
scopic  investigations.  In  the  first  system,  due  to  Hartley,  the 
wave-lengths  or  their  reciprocals,  the  frequencies,  are  plotted  as 
abscissae  and  the  thicknesses  of  the  absorbing  layers,  in  milli- 
meters, are  plotted  as  ordinates.  Such  curves  are  known  as 
curves  of  molecular  vibration.  The  second  system,  due  to  Baly 
and  Desch,  is  a  modification  of  that  developed  by  Hartley. 


Fig.  36. 

Baly  and  Desch  suggested  that  for  various  reasons  it  would  be 
more  advantageous,  if  instead  of  plotting  the  thicknesses  of  the 
absorbing  layers  as  ordinates,  the  logarithms  of  these  thicknesses 
be  plotted.  Both  methods  have  their  advantages  and  both  are 
used.  As  an  illustration  of  the  value  of  curves  of  molecular 
vibration  in  connection  with  questions  of  chemical  constitution, 
we  will  take  the  case  of  o-hydroxy-carbanil.  The  constitution 
of  this  substance  was  known  to  be  represented  by  one  of  the  two 
following  formulas:  — 

/°\  /°\ 

C6H/      ^>CO,     or    C6H</       V;  -  OH. 

NH  NX 

(a)  (b) 

Hartley  showed  that  the  absorption  spectra  of  the  two  ethyl 
derivatives,  the  lactam  and  lactim  ethers,   are  very  different. 


LIQUIDS 


143 


On  comparing  the  curves  of  molecular  vibration  for  the  three  sub- 
stances (Fig.  37),  it  is  apparent  that  the  curves  for  the  lactam 
ether  and  o-hydroxy-carbanil  bear  a  close  resemblance  to  each 
other,  while  the  curve  for  the  lactim  ether  is  very  different  from 
the  curves  for  the  other  two  substances.  The  constitution  of 


4000 


Oscillation  Frequencies 
4000 


4000 


3    4 
-5 

-4 

56789 

I 

/ 

2 

345 

6789 

1    2 

3    4 

56789         123 

1 

/ 

J 

1 

-3 

a 

3 

i 

H 

-2 

/ 

• 

-1 

/ 

/ 

i 

H 

L_ 

-4 
-3 
-2 
-1 

J 

\ 

J 

-A 

Lactam  Ether                 O-Hydroxy-CarJbanil                 Lactim  Ether 

Fig.  37. 

o-hydroxy-carbanil  must  then  be  very  similar  to  that  of  the 
lactam   ether.     The   formulas   of   the   ethyl   derivatives   of  the 
mother  substance  are  known  to  be  as  follows :  — 
.0,  O 

JO  C6H4<        >C  -  O.C2H6 


N  -  C2H5 
Lactam  ether 


Lactim  ether 


144  THEORETICAL  CHEMISTRY 

Hartley  concluded,  therefore,  that  formula  (a)  represents  the 
structure  of  the  molecule  of  o-hydroxy-carbanil. 

It  is  beyond  the  scope  of  this  book  to  discuss  at  greater  length 
the  bearing  of  absorption  spectra  upon  chemical  constitution;  but 
the  student  is  earnestly  advised  to  consult  some  book  *  treating  of 
this  important  subject  or  to  read  some  of  the  original  papers. 

Surface  Tension.  The  attraction  between  the  molecules  of  a 
liquid  manifests  itself  near  the  surface  where  the  molecules  are 
subject  to  an  unbalanced  internal  force.  The  condition  of  a 
liquid  near  its  surface  is  roughly  depicted  in  Fig.  38,  where  the 
dots  A,  B,  and  C  represent  molecules  and  the  circles  represent  the 
spheres  within  which  lie  all  of  the  other  molecules  which  exert  an 
appreciable  attraction  upon  A,  B,  and  C.  The  shaded  portions  rep- 


© 


Fig.  38. 


resent  those  molecules  whose  attractions  are  unbalanced.  These 
unbalanced  forces  will  evidently  tend  to  diminish  the  surface  to 
a  minimum  value.  That  is,  the  contraction  of  the  surface  of  a 
liquid  involves  the  expenditure  of  energy  by  the  liquid.  The 
surface  film  of  a  liquid  is  consequently  in  a  state  of  tension. 

Some  liquids  wet  the  walls  of  a  glass  capillary  tube  while  others  do 
not.  When  the  liquid  wets  the  tube,  the  surface  is  concave  and  the 
liquid  rises  in  the  tube;  on  the  other  hand,  when  the  liquid  does  not 
adhere,  the  surface  is  convex  and  the  liquid  is  depressed  in  the  tube. 
The  law  governing  the  elevation  or  depression  of  a  liquid  in  a 
capillary  tube  was  discovered  by  Jurin  and  may  be  stated  thus :  — 
The  elevation  or  depression  of  a  liquid  in  a  capillary  tube  is  inversely 
proportional  to  the  diameter  of  the  tube.  Let  Fig.  39  represent  a 

*  Relation  between  Chem.  Constitution  and  Phys.  Properties.    Samuel 
Smiles. 


LIQUIDS 


145 


capillary  tube  of  radius  r,  immersed  in  a  vessel  of  liquid  whose 
density  is  d,  and  let  the  elevation  of  the  liquid  in  the  capil- 
lary be  denoted  by  h.  Then  the  weight  of  the  column  of  liquid 
in  the  capillary  will  be  irr2hdg,  where  g  is  the  acceleration  due 
to  gravity.  The  force  sustaining  this  weight  is  2  irry  cos  0,  the 


Fig.  39. 

vertical  component,  of  the  force  due  to  the  tension  of  the  liquid 
surface  at  the  walls  of  the  tube,  7  being  the  surface  tension  and 
0  the  angle  of  contact  of  the  liquid  surface  with  the  walls  of 
the  tube. 


Therefore 


or 


Trr2hdg  =  2  irry  cos  0, 
hdgr 


7  = 


2cos0 


In  the  case  of  water  and  many  other  liquids  0  is  so  small  that  we 
may  write  0  =  0,  the  foregoing  expression  becoming 

7  =  1/2  hdgr. 

Thus  the  surface  tension  of  a  liquid  can  be  calculated  provided 
its  density  and  the  height  to  which  it  rises  in  a  previously  calibrated 
tube  is  known.  When  h  and  r  are  expressed  in  centimeters,  y 
will  be  expressed  in  dynes  per  centimeter  or  ergs  per  square  centi- 
meter. A  simple  form  of  apparatus  for  the  determination  of 
surface  tension  used  by  the  author  is  shown  in  Fig.  40.  A  capil- 
lary ,tube,  A,  of  uniform  bore  is  sealed  to  a  glass  rod,  E,  which  is 
held  in  position  in  the  test  tube,  B,  by  means  of  a  cork  stopper. 
A  short  right-angled  tube,  D,  and  a  thermometer,  F,  are  also 


146 


THEORETICAL  CHEMISTRY 


passed  through  the  same  cork  stopper.  The  liquid  whose  surface 
tension  is  to  be  measured  is  introduced  into  the  tube,  B,  the  cork 
inserted  and  the  tube  placed  inside 
of  the  larger  tube,  C,  containing  a 
liquid  of  known  boiling-point. 
When  the  thermometer,  F,  has 
become  stationary,  the  capillary 
elevation  of  the  liquid  is  measured  ( 
with  a  cathetometer.  The  tube, 
D,  permits  the  escape  of  vapor 
from  the  liquid  in  B  and  at  the 
same  time  insures  equality  of  pres- 
sure inside  and  outside  of  the  ap- 
paratus. The  spiral  tube,  G,  serves 
as  an  air  condenser,  preventing 
loss  of  vapor  from  the  liquid  in  the 
outer  tube.  The  surface  tension 
of  a  liquid  has  been  found  to  depend 
upon  the  nature  of  the  liquid  and 
also  upon  its  temperature. 

Surface  Tension  and  Molecular 
Weight.  In  1886,  Eotvos  *  showed 
that  the  surface  tension  multiplied 
by  the  two-thirds  power  of  the  mo- 
lecular weight  and  specific  volume 
is  a  function  of  the  absolute  tem- 
perature, or 

y(Mv)l  =  f(T), 

where  7  is  the  surface  tension,  M 


Fig.  40. 


the  molecular  weight,  v  the  specific  volume  or  reciprocal  of  the 
density,  and  T  the  absolute  temperature.  Ramsay  and  Shields  f 
modified  the  equation  of  Eotvos  as  follows :  — 

7  (Mv)l  =  k(tc-t-  6),  (1) 

tc  being  the  critical  temperature  of  the  liquid,  t  the  temperature 
of  the  experiment,  and  k  a  constant  independent  of  the  nature  of 

*  Wied.  Ann.,  27,  448  (1886). 

t  Zeit.  phys.  Chem.,  12,  431  (1893). 


LIQUIDS  147 

the  liquid.  The  physical  significance  of  the  two-thirds  power  of 
the  molecular  volume  has  been  explained  by  Ostwald  in  the  follow- 
ing manner  :  —  Assuming  the  molecules  to  be  spherical,  we  shall 
have  for  two  different  liquids,  the  proportion 

Vi:F2::ri8:r2», 

where  Vi  and  Vz  represent  the  volumes  and  ri  and  r2  the  radii  of 
their  respective  molecules.  Similarly  the  ratio  of  the  surfaces, 
Si  and  >S2,  of  the  molecules  in  terms  of  their  respective  radii,  will 
be 

Sn&iirtirf. 

From  these  two  proportions  it  follows  that  the  ratio  of  the  molec- 
ular surfaces  in  terms  of  the  molecular  volumes,  will  be 

Si:&::Fi*:Fs*. 

Making  use  of  the  value  of  M  as  determined  in  the  gaseous  state, 
Ramsay  and  Shields  found  the  value  of  k  for  a  large  number  of 
liquids  to  be  equal  to  2.12  ergs.  Among  the  liquids  for  which 
this  value  of  k  was  found  were  benzene,  carbon  tetrachloride, 
carbon  disulphide  and  phosphorus  trichloride.  For  certain  other 
liquids  such  as  water,  methyl  and  ethyl  alcohols  and  acetic  acid, 
k  was  found  to  have  values  much  smaller  than  2.12.  Ramsay 
and  Shields  attributed  these  abnormalities  to  an  increase  in  molec- 
ular weight  due  to  association,  and  suggested  that  the  degree  of 
association  might  be  calculated  from  the  equation 

xi  =  2.12/fc', 
or 


where  x  denotes  the  factor  of  association,  and  k'  is  the  value  of 
the  constant  for  the  associated  liquid  in  equation  (1).  It  was 
further  pointed  out  by  Ramsay  and  Shields  that  equation  (1) 
affords  a  means  of  calculating  the  molecular  weight  of  a  pure 
liquid,  provided  we  assume  that  for  a  non-associated  liquid  the 
mean  value  of  k  is  2.12.  Since  it  is  not  an  easy  matter  to  deter- 
mine the  critical  temperature  with  accuracy,  Ramsay  and  Shields 
made  use  of  a  differential  method,  and  thus  eliminated  te  from 
equation  (1).  If  the  surface  tension  of  a  liquid  be  measured  at 


148  THEORETICAL  CHEMISTRY 

two  temperatures  h~and  k,  and  the  corresponding  densities  are  di 
and  dz,  we  shall  have 

71  (M/di)i  =  k(te-ti-  6),  (3) 
and 

72  (M/d,)i  =  k(tc-U-  6).  (4) 
Subtracting  equation  (4)  from  equation  (3),  we  obtain 


or  solving  equation  (5)  for  M,  we  have 


The  method  of  Ramsay  and  Shields  is  the  best  known  method  for 
the  determination  of  the  molecular  weight  of  a  pure  liquid.  If 
M  is  known  to  be  the  same  in  the  liquid  and  gaseous  states, 
orj  in  other  words,  if  k  is  independent  of  the  temperature,  even 
though  its  value  is  not  exactly  2.12,  the  critical  temperature  of 
the  liquid  can  be  calculated  by  means  of  equation  (1).  In  order 
that  the^correct  value  of  the  critical  temperature  may  be  obtained, 
Ramsay  and  Shields  found  it  necessary  to  use  the  specific  value 
of  k  for  the  liquid  whose  critical  temperature  is  sought.  As  an 
illustration  of  the  method  of  calculation,  the  following  example  is 
taken  from  the  work  of  Ramsay  and  Shields. 
For  carbon  disulphide, 

7  at  19°.4  =  33.58  7  at  46°.  1  =  29.41 

d  at  19°.4  =  1.264  d  at  46°.  1  =  1.223. 

We  have  then  for  7  (M/d)l,  at  the  two  temperatures, 

(76/1.264)1  X  33.58  =  515.4, 
and 

(76/1.223)1  X  29.41  =  461.4. 
Substituting  in  the  equation 

71  (MM)!  -  72(M/d2)f 

*> 


tz  —  ti 
we  have, 

515.4  -  461.4 
46.1  -  19.4 


=  2.022. 


LIQUIDS  149 

This  value  of  k  is  so  nearly  equal  to  the  mean  value,  2.12,  that  we 
assume  M  to  be  the  same  in  the  liquid  and  gaseous  states,  and 
therefore  we  may  substitute  in  equation  (1)  and  calculate  the 
critical  temperature  of  carbon  disulphide  thus, 

7  (M/d)l  =  k  (tc  -  t-  6), 
or  solving  for  tc)  we  have 


tc  =  fa         -  +  t  +  0. 

Substituting  the  data  given  above,  in  the  preceding  equation,  we 
obtain 

tc  =  515.4/2.022  +  6  +  19.4, 
or  tc  =  280°.3  C. 

Surface  Tension  and  Drop-Weight.  Morgan  and  his  co- 
workers,*  from  measurements  of  the  volumes  of  a  single  drop 
falling  from  the  carefully-ground  tip  of  a  capillary  tube,  have 
shown  that  the  weight  of  the  falling  drop,  from  such  a  tip  can  be 
used  in  place  of  the  surface  tension  in  the  equation  of  Ramsay 
and  Shields  for  the  calculation  of  molecular  weights  and  critical 
temperatures.  The  modified  equation  may  be  written  thus  :  — 

_  ln 


where  wi  amd  wz  are  the  respective  weights  of  the  falling  drop 
at  the  temperatures,  ti  and  t2.  The  value  of  k  obviously  depends 
upon  the  tip  employed. 

The  results  obtained  by  the  drop-weight  method  have  been 
shown  to  be  more  trustworthy  than  those  obtained  by  the  method 
of  capillary  elevation.  Morgan  has  further  pointed  out  that 
when  the  experimental  data  are  substituted  in  the  preceding 
formula,  the  magnification  of  the  experimental  errors  is  appreci- 
ably greater  than  when  use  is  made  of  the  original  formula, 
w  (M/d)t  =  k(tc-t-  6). 

Morgan  recommends  therefore  that  this  formula  be  used  for 
the  determination  of  molecular  weights.     After  having  calibrated 

*  Jour.  Am.  Chem,  Soc.,  30,  360  (1908);  30,  1055  (1908). 


150  THEORETICAL  CHEMISTRY 

a  particular  tip  with  pure  benzene  (a  liquid  which  is  known  to  be 
non-associated),  and  thus  ascertaining  the  value  of  k,  the  drop- 
weights  at  several  different  temperatures  are  determined.  If 
we  assume  M  to  have  the  same  value  in  the  liquid  and  gaseous 
states,  the  value  of  tc  can  be  computed  by  substituting  the  experi- 
mental data  in  the  preceding  equation.  If  at  the  different  tem- 
peratures at  which  drop-weights  are  determined,  the  same  value 
of  tc  is  obtained,  then  we  may  infer  that  the  liquid  is  non-asso- 
ciated and,  therefore,  that  the  assumption  made  as  to  the  value  of 
M  is  confirmed.  It  is  a  singular  fact  that  the  calculated  value 
of  tc  for  some  liquids  does  not  agree  with  the  experimental  value, 
although  it  remains  constant  throughout  an  extended  range  of 
temperatures.  Morgan  considers  a  constant  value  of  tc  to  be  an 
indication  of  non-association,  even  if  the  value  is  fictitious.  In 
this  method  the  constancy  of  the  calculated  value  of  the  critical 
temperature  becomes  the  criterion  of  molecular  association,  and 
thus  affords  a  means  of  determining  whether  the  molecular  weight 
in  the  liquid  state  is  identical  with  that  in  the  gaseous  state.  The 
values  of  tc  calculated  from  the  drop-weights  of  an  associated 
liquid  become  steadily  smaller  as  the  temperature  increases.  A 
large  number  of  liquids  have  been  studied  by  this  method,  and 
the  results  indicate  that  many  of  the  substances  which  were  con- 
sidered to  be  associated  by  Ramsay  and  Shields  are  in  reality 
non-associated  ;  in  fact,  it  appears  from  the  work  of  Morgan  that 
association  is  much  less  common  among  liquids  than  has  hitherto 
been  supposed. 

Dielectric  Constants.  In  1837,  Faraday  discovered  that  the 
attraction  or  repulsion  between  two  electric  charges  varies  with 
the  nature  of  the  intervening  medium  or  dielectric.  If  qi  and  q2 
represent  two  charges  which  are  separated  by  a  distance  r,  the 
force  of  attraction  or  repulsion,  /,  is  given  by  the  equation 


D'  r* 


where  D  is  a  specific  property  of  the  medium  known  as  the  dielectric 
constant.  The  dielectric  constant  of  air  is  taken  as  unity.  Var- 
ious methods  have  been  devised  for  the  experimental  determination 


LIQUIDS 


151 


of  the  dielectric  constant,  but  the  scope  of  this  book  forbids 
even  a  brief  description  of  the  apparatus  or  an  outline  of  the 
processes  of  measurement.  For  a  description  of  these  methods 
the  student  is  referred  to  any  one  of  the  more  complete  physico- 
chemical  laboratory  manuals,  or  to  the  original  communications 
of  Nernst  *  and  Drude.f 

The  values  of  the  dielectric  constants  for  some  of  the  more 
common  solvents  are  given  in  the  accompanying  table. 

DIELECTRIC  CONSTANTS  AT  18°  C. 


Substance. 

D 

Hydrogen  dioxide  

92  8 

Water  

77  0 

Formic  acid  

63  0 

Methyl  alcohol  

33  7 

Ethyl  alcohol  

25.9 

Ammonia,  liquid  

22  0 

Chloroform  

5  0 

Ether  

4.4 

Carbon  disulphide  

2.6 

Benzene  

2.3 

The  importance  of  this  property  of  liquids  will  become  more 
apparent  in  subsequent  chapters,  especially  in  those  devoted  to 
3lectrochemistry. 

PROBLEMS. 

1.  It  is  desired  to  compare  the  molecular  volumes  of  alcohol  and  ether. 
If  the  molecular  volume  of  ether  is  determined  at  20°  C.,  at  what  temper- 
ature must  the  molecular  volume  of  alcohol  be  determined?  The  boiling 
points  of  alcohol  and  ether  are  78°  and  35°  respectively.      Ans.   61  ^C. 

2.  A  volume  of  50  liters  of  air  in  passing  through  a  liquid  at  22°  C. 
causes  the  evaporation  of  5  grams  of  substance,  the  molecular  weight  of 
which  is  100.     What  is  the  vapor  pressure  of  the  liquid  in  grams  per 
square  centimeter?  Ans.  25. 

*  Zeit.  phys.  Chem.,  14,  622  (1894). 
t  Ibid.,  23,  267  (1897). 


152  THEORETICAL  CHEMISTRY 

3.  The  boiling-point  of  ethyl   propionate  is  98°.7  C.  and  its  heat  of 
vaporization  is  77.1  calories.     Calculate  its  molecular  weight. 

4.  The  heat  of  vaporization  of  liquid   ammonia  at  its  boiling-point, 
under  atmospheric  pressure  (—  33°. 5  C.)  is  341  calories.     Is  liquid  am- 
monia associated? 

5.  Calculate  the  molecular  volume  of  ethyl  butyrate.     The  molecular 
volume  determined  by  experiment  is  149.1. 

6.  For  propionic  acid,  d  =  1.0158  and  no  =  1.3953.     Calculate  the 
molecular  refraction  by  the  formula  of  Lorenz-Lorentz  and  compare  the 
value  so  obtained  with  that  derived  from  the  atomic  refractions   of 
the  constituent  elements. 

7.  The  density  of  ether  is  0.7208,  of  ethyl  alcohol,  0.7935  and  of  a 
mixture  of  ether  and  alcohol  containing  p  per  cent  of  the  latter,  0.7389. 
At  20°  C.  the  refractive  indices  for  sodium  light  are,  for  ether,  1.3536,  for 
alcohol,  1.3619,  and  for  the  mixture,  1.3572.     Calculate  the  value  of  p, 
using  the  Gladstone  and  Dale  formula.  Ans.  20.81. 

8.  At  20°  C.  the  density  of  chloroform  is  1.4823  and  the  refractive 
index  for  the  D-line  is  1.4472.     Given  the  atomic  refractivities  of  carbon 
and   hydrogen,   calculate   that   of  chlorine,   using   the  Lorenz-Lorentz 
formula.  Ans.   5.999. 

9.  Calculate  the  surface  tension  of  benzene  in  dynes  per  centimeter  t 
the  radius  of  the  capillary  tube  being  0.01843  cm.,  the  density  of  the 
liquid,  0.85,  and  the  height  to  which  it  rises  in  the  capillary,  3.213  cm. 

Ans.  24.71  dynes/cm. 

10.  Find  the  molecular  weight  of  benzene,  the  surface  tension  at 
46°  C.  being  24.71  dynes  per  centimeter,  its  critical  temperature,  288°.5  C., 
its  density,  0.85  and  the  value  of  k  =  2.12.  Ans.  77.7. 

11.  At  14°.8  C.  acetyl  chloride  (density  =  1.124)  ascends  to  a  height 
of  3.28  cm.  in  a  capillary  tube  the  radius  of  which  is  0.01425  cm.     At 
46°.2  C.  in  the  same  tube  the  elevation  is  2.85  cm.  and  the  density 
=  1.064.     Calculate  the  critical  temperature  of  acetyl  chloride. 

Ans.  234°.6C. 

12.  From  a  certain  tip  the  weights  of  a  falling  drop  of  benzene  are 
35.329  milligrams  (temp.  =  11°.4,  density  =  0.888)  and    26.530    milli- 
grams (temp.  =  68°.5,  density  =  0.827).    The  molecular  weight  is  the 
same  in  the  liquid  and  gaseous  states.     Calculate  the  critical  temper- 
ature of  benzene.  Ans.   286°.  1C. 


CHAPTER  VIII. 
SOLIDS. 

General  Properties  of  Solids.  Solids  differ  from  gases  and 
liquids  in  possessing  definite,  individual  forms.  Matter  in  the 
solid  state  is  capable  of  resisting  considerable  shearing  and  tensile 
stresses.  In  terms  of  the  kinetic  theory  of  matter,  the  mutual 
attractive  forces  exerted  by  the  molecules  of  solids  must  be  re- 
garded as  superior  to  the  attractive  forces  between  the  molecules 
of  gases  and  liquids.  With  one  or  two  exceptions  all  solids  ex- 
pand when  heated,  but  there  is  no  simple  law  expressing  the  relation 
between  the  increment  of  volume  and  the  temperature.  Rigidity  is 
another  characteristic  property  of  solids,  it  being  much  more  ap- 
parent in  some  than  in  others.  Many  solids  are  constantly  under- 
going a  process  of  transformation  into  the  gaseous  state  at  their 
free  surfaces,  such  a  change  being  known  as  sublimation.  Just 
as  when  a  gas  is  sufficiently  cooled  it  passes  into  the  liquid  state, 
so  on  cooling  a  liquid  below  a  certain  temperature,  it  passes  into 
the  solid  state.  The  reverse  transformations  are  also  possible,  a 
solid  being  liquefied  when  sufficiently  heated,  and  the  resulting 
liquid  completely  vaporized  if  the  heating  be  continued.  Heat 
energy  is  required  to  effect  transition  from  the  solid  to  the  liquid 
state,  just  as  heat  energy  is  required  to  effect  transition  from  the 
liquid  to  the  gaseous  state. 

Obviously  a  substance  in  the  solid  state  contains  less  energy 
than  it  does  in  the  liquid  state.  The  number  of  calories  required 
to  melt  1  gram  of  a  solid  substance  is  called  its  heat  of  fusion. 
It  is  often  difficult  to  decide  whether  a  substance  should  be  classi- 
fied as  a  solid  or  as  a  liquid.  For  example  the  behavior  of  certain 
amorphous  substances  such  as  pitch,  amber  and  glass,  is  similar 
to  that  of  a  very  viscous,  inelastic  liquid.  Solids  are  generally 
classified  as  crystalline  or  amorphous.  In  crystalline  solids  the 
molecules  are  supposed  to  be  arranged  in  some  definite  order,  this 

153 


154  THEORETICAL  CHEMISTRY 

arrangement  manifesting  itself  in  the  crystal  form.  An  amor- 
phous solid  on  the  other  hand  may  be  considered  as  a  liquid 
possessing  great  viscosity  and  small  elasticity.  The  physical 
properties  of  amorphous  solids  have  the  same  values  in  all  direc- 
tions, whereas  in  crystalline  solids  the  values  of  these  properties 
may  be  different  in  different  directions.  When  an  amorphous 
solid  is  heated  it  gradually  softens  and  eventually  acquires  the 
properties  characteristic  of  a  liquid,  but  during  the  process  of  heat- 
ing there  is  no  definite  point  of  transition  from  the  solid  to  the 
liquid  state.  On  the  other  hand,  when  a  crystalline  solid  is 
heated  there  is  a  sharp  change  from  one  state  to  the  other  at  a 
definite  temperature,  this  temperature  being  termed  the  melting- 
point. 

Crystallography.  The  study  of  the  definite  geometrical  forms 
assumed  by  crystalline  solids  is  termed  crystallography.  The 
number  of  crystalline  forms  known  is  exceedingly  large,  but  it  is 
possible  to  reduce  the  many  varieties  to  a  few  classes  or  systems 
by  referring  their  principal  elements  —  the  planes  —  to  definite 
lines  called  axes.  These  axes  are  so  drawn  within  the  crystal  that 
the  crystal  surfaces  are  symmetrically  arranged  about  them. 
This  system  of  classification  was  proposed  by  Weiss  in  1809. 
He  showed  that  notwithstanding  the  multiplicity  of  crystal  forms 
encountered  in  nature,  it  is  possible  to  consider  them  as  belonging 
to  one  of  six  systems  of  crystallization. 

The  six  systems  of  Weiss  are  as  follows:  — 

1.  The  Regular  System.    Three   axes  of  equal  length,  inter- 
secting each  other  at  right  angles  (Fig.  41). 

2.  The  Tetragonal  System.    Two  axes  of  equal  length  and  the 
third  axis  either  longer  or  shorter,  all  three  axes  intersecting  at 
right  angles  (Fig.  42). 

3.  The  Hexagonal  System.    Three  axes  of  equal  length,  all  in 
the  same  plane  and  intersecting  at  angles  of  60°,  and  a  fourth  axis, 
either  longer  or  shorter  and  perpendicular  to  the  plane  of  the 
other  three  (Fig.  43). 

4.  The  Rhombic  System.    Three  axes  of  unequal  length,  all 
intersecting  each  other  at  right  angles  (Fig.  44). 


SOLIDS 


155 


5.  The  Monodinic  System.    Three  axes  of  unequal  length,  two 
of  which  intersect  at  right  angles,  while  the  third  axis  is  per- 
pendicular to  one  and  not  to  the  other  (Fig.  45). 

6.  The  Tridinic  System.    Three  axes  of  unequal  length  no  two 
of  which  intersect  at  right  angles  (Fig.  46). 

The  position  of  a  plane  in  space  is  determined  by  three  points 
in  a  system  of  coordinates,  and  consequently  the  position  of  the 


a 
Fig.  41. 


Fig.  42. 


Fig.  43. 


Fig.  44. 


Fig.  45. 


Fig.  46. 


face  of  a  crystal  is  likewise  determined  by  its  points  of  inter- 
section with  the  three  axes,  or  by  the  distances  from  the  origin 
of  the  system  of  coordinates  at  which  the  plane  of  the  crystal 
face  intersects  the  three  axes.  These  distances  are  called  the 
parameters  of  the  plane. 

The  fundamental  law  of  crystallography  discovered  by  Steno 


156  THEORETICAL  CHEMISTRY 

in  1669  may  be  stated  thus:  —  The  angle  between  two  given  crystal 
faces  is  always  the  same  for  the  same  substance.  The  fact  that 
every  crystalline  substance  is  characterized  by  a  constant  inter- 
facial  angle,  affords  a  valuable  means  of  identification  which  is 
used  by  both  chemists  and  mineralogists.  The  instrument  em- 
ployed for  the  measurement  of  the  interfacial  angles  of  crystals  is 
called  a  goniometer.  The  crystal  to  be  measured  is  mounted  at 
the  center  of  the  graduated  circular  table  of  the  goniometer,  and 
the  image  of  an  illuminated  slit,  reflected  from  one  surface  of  the 
crystal,  is  brought  into  coincidence  with  the  cross-wires  in  the  eye- 
piece of  the  telescope.  The  table  is  then  turned  until  the  image 
of  the  slit,  reflected  from  the  adjacent  face  of  the  crystal,  coincides 
with  the  cross-wires.  The  interfacial  angle  of  the  crystal  is  de- 
termined by  the  number  of  degrees  through  which  the  table  has 
been  turned. 

Properties  of  Crystals.  The  properties  of  all  crystals,  except 
those  belonging  to  the  regular  system,  exhibit  differences  depend- 
ent upon  the  direction  in  which  the  particular  measurements 
are  made.  Thus  the  elasticity,  the  thermal  and  electrical  con- 
ductivities, and  in  fact  all  of  the  physical  properties  of  crystals 
which  do  not  belong  to  the  regular  system,  have  different  values 
in  different  directions.  Crystals  whose  physical  properties  have 
the  same  values  in  all  directions  are  termed  isotropic,  while  those 
in  which  the  values  are  dependent  upon  the  direction  in  which 
the  measurements  are  made,  are  called  anisotropic.  Crystals 
belonging  to  the  regular  system,  and  amorphous  substances  are 
isotropic.  Certain  amorphous  substances,  such  as  glass,  which 
are  normally  isotropic,  may  become  anisotropic  when  subjected 
to  tension  or  compression.  The  phenomenon  of  double  refrac- 
tion observed  in  all  crystals,  except  those  belonging  to  the  regular 
system,  is  due  to  their  anisotropic  character.  Crystals  belonging 
to  the  tetragonal  and  hexagonal  systems  resemble  each  other  in 
one  respect,  viz.:  that  in  all  of  them  there  is  one  direction,  called 
the  optic  axis,  or  axis  of  double  refraction  (coincident  with  the  prin- 
cipal crystallographic  axis),  along  which  a  ray  of  light  is  singly 
refracted,  while  in  all  other  directions  it  is  doubly  refracted.  In 
crystals  belonging  to  the  rhombic,  monoclinic,  and  triclinic  systems, 


SOLIDS  157 

there  are  always  two  directions  along  which  a  ray  of  light  is  singly 
refracted.  A  crystal  of  Iceland  spar  (CaCO3)  affords  a  beauti- 
ful illustration  of  double  refraction.  On  placing  a  rhomb  of  this 
substance  over  a  piece  of  white  paper  on  which  there  is  an  ink 
spot,  two  spots  will  be  seen.  On  turning  the  crystal,  one  spot  will 
remain  stationary  while  the  other  spot  will  revolve  about  it.  This 
property  of  Iceland  spar  is  utilized  in  the  construction  of  Nicol 
prisms  for  polariscopes. 

The  examination  of  sections  of  anisotropic  crystals  in  a  polari- 
scope  between  crossed  Nicol  prisms,  reveals  something  as  to  their 
crystal  form.  As  has  been  stated,  crystals  of  the  tetragonal  and 
hexagonal  systems  are  uniaxial.  If  a  section  is  cut  from  such  a 
crystal  perpendicular  to  the  optic  axis,  and  this  is  placed  between 
the  crossed  Nicol  prisms  of  a  polariscope,  in  a  convergent  beam 
of  white  light,  a  dark  cross  and  concentric,  spectral-colored  circles 
will  be  observed,  Fig.  47.  Upon  turning  the  analyzer  through 
90°  the  colors  of  the  circles  will  change  to  the  respective  comple- 
mentary colors  and  the  dark  cross  will  become  light.  Crystals 
of  the  rhombic,  monoclinic,  and  triclinic  systems  are  biaxial.  If 
a  section  of  a  biaxial  crystal,  cut  perpendicular  to  the  line  bisect- 
ing the  angle  between  the  two  axes,  be  placed  in  the  polariscope 


Fig.  46.  Fig.  47. 

and  examined  as  in  the  preceding  case,  a  series  of  concentric 
spectral-colored  lemniscates  surrounding  two  dark  centers  and 
pierced  by  dark,  hyperbolic  brushes,  will  be  observed,  as  shown  in 
Fig.  48.  On  rotating  the  analyzer,  the  colors  will  change  to  the 
corresponding  complementary  colors,  as  in  the  case  of  uniaxial 
crystals.  The  appearance  of  these  figures  is  so  varied  and  cha- 
racteristic as  to  furnish,  in  many  cases,  a  very  satisfactory  means 
of  identifying  anisotropic  crystals. 


158  THEORETICAL  CHEMISTRY 

x 

Etch  Figures.  The  solubility  of  crystals  has  been  shown  to 
be  different  in  different  directions.  Thus,  if  the  surface  of  a  crys- 
talline substance  be  highly  polished  and  then  treated  for  a  short 
time  with  a  suitable  solvent,  faint  patterns,  known  as  etch  figures, 
will  appear  as  a  result  of  the  inequality  of  the  rate  of  solution  in 
different  directions.  When  these  figures  are  examined  under  the 
microscope  the  crystal-form  can  generally  be  determined.  The 
examination  of  etch  figures  has  come  to  be  of  prime  importance 
to  the  metallographer.  Thus,  when  an  appropriate  solvent  is 
applied  to  the  polished  surface  of  an  alloy,  not  only  is  the  crystal 
form  revealed  by  the  etch  figures,  but  also  the  presence  of  various 
chemical  compounds  may  be  recognized.  By  a  careful  study 
of  the  etch  figures  developed  on  the  surface  of  highly  polished 
steel,  the  metallographer  may  gather  important  information  as  to 
its  previous  history,  especially  its  heat  treatment. 

Crystal  Form  and  Chemical  Composition.  From  the  pre- 
ceding paragraphs  it  might  be  inferred  that  the  same  substance 
always  assumes  the  same  crystal  form.  While  this  is  true 
in  general,  there  are  some  substances  which  appear  in  several 
different  crystal  forms.  This  phenomenon  is  termed  polymor- 
phism. 

Calcium  carbonate  is  an  example  of  a  substance  crystallizing  in 
more  than  one  form.  As  calcite,  it  crystallizes  in  the  hexagonal 
system,  while  as  aragonite,  it  crystallizes  in  the  rhombic  system. 
Such  a  substance  is  said  to  be  dimorphous.  Of  the  several  factors 
controlling  polymorphism,  temperature  is  the  most  important. 
Thus  sulphur  crystallizes  at  temperatures  above  95°.6  in  the  mon- 
oclinic  system,  while  at  lower  temperatures  it  assumes  the 
rhombic  form.  The  temperature  at  which  it  changes  from  one 
form  into  the  other  is  termed  its  transition  temperature.  As  has 
been  mentioned  in  an  earlier  chapter  (p.  14),  some  substances 
may  crystallize  in  the  same  form,  the  characteristic  interfacial 
angles  being  nearly  identical.  Such  substances  are  said  to  be  iso- 
morphous.  This  phenomenon,  discovered  by  Mitscherlich,  has 
been  of  great  use  in  connection  with  the  earlier  investigations  on 
atomic  weights,  as  has  already  been  pointed  out. 

There  can  be  little  doubt  as  to  the  existence  of  an  intimate 


SOLIDS  159 

connection  between  crystalline  form  and  chemical  composition. 
Ever  since  the  early  part  of  the  nineteenth  century,  when  Haiiy 
established  the  science  of  crystallography,  various  attempts  have 
been  made  by  chemists  and  crystallographers  to  connect  crystal- 
line form  with  chemical  constitution.  In  1906,  Barlow  and  Pope  * 
made  a  most  notable  contribution  to  the  theories  concerning  the 
relation  between  crystalline  form  and  chemical  constitution. 
Their  ideas  may  be  summarized  as  follows :  —  If  each  atom  be 
considered  as  appropriating  a  certain  space,  called  its  sphere  of 
atomic  influence,  then  (1)  the  spheres  of  atomic  influence  are  so 
arranged  as  to  occupy  the  smallest  possible  volume  in  every  crystal; 
(2)  the  volumes  of  the  spheres  of  atomic  influence  in  any  substance 
are  proportional  to  the  valences  of  the  constituent  atoms;  (3)  the 
volumes  of  the  spheres  of  influence  of  the  atoms  of  different  elements 
of  the  same  valence  are  nearly  equal,  any  variation  being  in  har- 
mony with  their  relations  in  the  periodic  system.  Barlow  and  Pope 
have  shown  that  the  general  agreement  between  theory  and 
observation  is  most  satisfactory,  a  particularly  strong  argument 
in  favor  of  this  theory  being  the  very  plausible  explanation  which 
it  furnishes  for  a  large  number  of  crystallographic  facts.  It  is 
without  doubt  the  best  working  hypothesis  which  has  yet  been 
offered  for  the  investigation  of  the  dependence  of  crystalline  form 
upon  a  definite  chemical  constitution. 

Compressibilities  of  the  Solid  Elements.  A  series  of  careful 
measurements  of  the  compressibilities  of  the  elements  by  T.  W. 
Richards  and  his  collaborators,!  has  revealed  the  fact  that  com- 
pressibility is  a  periodic  function  of  atomic  weight.  Richards 
has  advanced  some  interesting  suggestions  as  to  the  importance 
of  compressibility  in  connection  with  intermolecular  cohesion 
and  atomic  volume.  In  fact,  Richards'  theory  of  compressible 
atoms  may  be  regarded  as  a  valuable  supplement  to  the  the- 
ory of  Barlow  and  Pope  and,  taken  together,  these  two  theories 
constitute  a  rational  basis  for  the  science  of  chemical  crystallog- 
raphy. 

X-Rays  and  Crystal  Structure.     In  1912,  Laue  pointed  out 

*  Jour.  Chem.  Soc.,  91,  1150  (1907). 

f  Zeit.  phys.  Chem.,  61,  77,  100,  171,  183  (1908). 


160  THEORETICAL  CHEMISTRY 

that  the  regularly  arranged  atoms  or  molecules  of  a  crystal  should 
act  as  a  three-dimensional  diffraction  grating  toward  the  X-rays. 
He  showed  mathematically  that  on  traversing  a  thin  section  of  a 
crystal,  a  pencil  of  X-rays  should  give  rise  to  a  diffraction  pattern 
arranged  symmetrically  round  the  primary  beam  as  a  center.  A 
photographic  plate  placed  perpendicular  to  the  path  of  the  rays 
and  behind  the  crystal  should  reveal,  on  development,  a  central 
spot  due  to  the  action  of  the  primary  rays,  and  a  series  of  symmetri- 
cally grouped  spots  due  to  the  diffracted  rays. 

Laue's  predictions  were  verified  experimentally  by  Friedrich 
and  Knipping,  who  obtained  numerous  plates  showing  a  vari- 
ety of  geometrical  patterns  corresponding  to  the  structural  dif- 
ferences of  the  crystals  examined.  The  analysis  of  the  Laue 
diffraction  patterns,  while  furnishing  valuable  information  as  to 
the  internal  structure  of  crystals,  is  nevertheless  extremely 
complex. 

W.  H.  Bragg  and  his  son  W.  L.  Bragg  *  have  devised  an  X-ray 
spectrometer  in  which  use  is  made  of  the  fact  that  the  regularly 
spaced  atoms  of  a  crystal  reflect  the  X-rays  in  much  the  same  way 
that  light  is  reflected  (diffracted)  by  a  plane  grating.  By  observing 
the  angles  of  reflection  from  the  different  faces  of  a  crystal  for  an 
incident  radiation  of  known  wave-length,  it  is  an  easy  matter  to 
calculate  the  distances  between  the  atoms  of  the  crystal  which 
function  as  diffraction  centers. 

By  means  of  the  X-ray  spectrometer  the  internal  structure  of  a 
number  of  crystals  has  been  determined.  One  of  the  most  in- 
teresting results  to  the  chemist  is  that  obtained  with  the  diamond. 
The  X-ray  spectra  of  the  diamond  reveal  the  fact  that  each  carbon 
atom  is  situated  at  the  center  of  a  regular  tetrahedron  formed  by 
four  other  carbon  atoms. 

In  commenting  on  this  method  of  studying  crystal  structure, 
W.  H.  Bragg  says:  —  " Instead  of  guessing  the  internal  arrange- 
ment of  the  atoms  from  the  outward  form  assumed  by  the  crystal, 
we  find  ourselves  able  to  measure  the  actual  distances  from  atom 
to  atom  and  to  draw  a  diagram  as  if  we  were  making  a  plan  of  a 
building." 
*  See  "X-Rays  and  Crystal  Structure"  by  W.  H.  Bragg  and  W.  L.  Bragg. 


SOLIDS  161 

Heat  Capacity  of  Solids.  Recent  investigations  of  specific 
heats  of  solids  at  extremely  low  temperatures  have  resulted  in 
the  formulation  of  several  interesting  relationships  between  heat 
capacity  and  temperature. 

At  ordinary  temperatures  the  molecules  of  a  crystalline  solid 
may  be  assumed  to  be  in  a  state  of  violent,  unordered  motion.  As 
the  temperature  is  lowered,  the  amplitude  of  the  molecular  oscil- 
lations steadily  diminishes  until  finally,  at  the  absolute  zero,  there 
is  in  all  probability  a  complete  cessation  of  motion.  In  the  neigh- 
borhood of  the  absolute  zero,  where  the  amplitude  of  the  molecular 
oscillations  is  negligible,  a  crystalline  solid  may  be  assumed  to 
possess  the  properties  characteristic  of  a  perfectly  elastic  body. 
In  other  words,  the  crystalline  forces  holding  the  molecules  to- 
gether would  preponderate  over  the  feeble  thermal  forces  tending 
to  initiate  molecular  oscillations  within  the  solid.  Under  these 
conditions  the  solid  as  a  whole  would  exhibit  the  same  behavior  as 
a  single  molecule,  that  is  to  say,  the  solid  would  function  as  a 
perfectly  elastic  body. 

On  this  assumption  Debye  *  has  derived  the  following  equation 
expressing  the  heat  capacity  of  a  solid,  Cv,  in  terms  of  its  absolute 
temperature  T, 

TS 
Cv  =  71.903  • 

In  this  equation  6  is  a  constant  characteristic  of  each  solid  and  has 
the  same  dimensions  as  T.  The  value  of  6  varies  between  the 
limits  6  =  50  for  calcium  and  0  =  1840  for  carbon.  The  agree- 
ment between  the  observed  and  calculated  values  of  Cv  has  been 
found  to  be  excellent  up  to  T  =  9/12. 

When  this  latter  temperature  is  exceeded,  the  molecules  of  the 
solid  begin  to  absorb  more  and  more  heat  energy  and  to  vibrate 
independently  about  their  centers  of  oscillation.  The  failure  of 
Debye's  equation  is  to  be  expected  under  these  conditions  since 
the  solid  is  no  longer  behaving  as  one  large  molecule.  Obviously 
the  lighter  the  molecules  and  the  greater  the  crystalline  forces 
within  the  solid,  the  higher  must  the  temperature  become  before 

*  Ann.  Physik.,  39,  789  (1912). 


162  THEORETICAL  CHEMISTRY 

the  individual  molecules  can  acquire  appreciable  kinetic  energy. 
This  is  apparent  from  the  familiar  dynamical  principle,  that  the 
kinetic  energy  of  a  vibrating  particle  is  proportional  to  its  mass 
and  to  the  square  of  its  vibration  frequency.  In  the  case  of  lead, 
which  is  a  soft,  malleable  solid  with  a  relatively  low  melting  point, 
it  is  reasonable  to  infer  that  the  crystalline  forces  are  feeble,  and 
consequently  we  should  expect  that  molecular  and  atomic  vibra- 
tions would  be  set  up  at  quite  low  temperatures.  Furthermore, 
since  the  atoms  of  lead  are  extremely  heavy,  their  kinetic  energy 
must  be  great.  The  correctness  of  these  conclusions  is  confirmed 
by  the  fact  that  the  Debye  equation  when  applied  to  lead  has  been 
found  to  hold  only  over  a  very  short  range  of  temperature.  On 
the  other  hand,  the  equation  has  been  found  to  hold  for  the 
diamond  up  to  a  temperature  of  about  200°  absolute.  In  this 
case  we  have  a  solid  in  which  the  crystalline  forces  are  extremely 
powerful  and  in  which  the  atoms  are  relatively  light.  A  fairly 
high  temperature  must  be  attained  before  the  energy  absorbed 
by  the  individual  atoms  of  the  diamond  acquires  appreciable 
magnitude. 

The  absorption  of  energy  by  the  vibrating  molecules  continues 
to  increase  as  the  temperature  is  raised  until  ultimately,  at  the 
melting  point  of  the  solid,  the  crystalline  forces  become  negligible. 
As  this  temperature  is  approached  therefore,  the  intermolecu- 
lar  restraint  becomes  less  and  less  and  the  mean  kinetic  energy 
of  the  molecules  approaches  that  of  the  molecules  of  the  molten 
solid. 

As  has  already  been  stated  in  Chapter  I  (p.  11),  Dulong  and 
Petit,  in  1819,  discovered  the  interesting  fact  that  the  atomic  heats 
of  the  solid  elements  have  a  constant  value  of  6.5.  The  importance 
of  this  generalization  in  connection  with  the  verification  of  atomic 
weights  has  already  been  pointed  out.  Quite  recently,  Lewis  * 
has  directed  attention  to  the  fact  that  it  is  much  more  rational  to 
calculate  the  atomic  heat  of  an  element  from  the  specific  heat  at 
constant  volume  rather  than  from  the  specific  heat  at  constant 
pressure.  While  it  is  impossible  to  measure  the  specific  heat  at 
constant  volume,  its  value  may  be  derived  from  the  specific  heat 
*  Jour.  Am.  Chem.  Soc.,  29,  1165  (1907). 


SOLIDS  163 

at  constant  pressure  by  an  application  of  the  laws  of  thermo- 
dynamics. Thus,  Lewis  has  obtained  the  formula 

TaW 
Lp~         "41.780' 

where  T  denotes  the  absolute  temperature,  a  the  coefficient  of 
expansion,  0  the  coefficient  of  compressibility,  Cp  and  Cv  the 
atomic  specific  heats  at  constant  pressure  and  constant  volume 
respectively  and  V  the  atomic  volume.  By  means  of  this  equa- 
tion, Lewis  has  established  the  following  generalization:  —  Within 
the  limits  of  experimental  error,  the  atomic  heat  at  constant  volume, 
at  20°  C.,  is  the  same  for  all  solid  elements  whose  atomic  weights  are 
greater  than  that  of  potassium,  and  is  equal  to  5.9.  In  the  case  of  a 
solid  having  a  high  melting  point,  the  violent  agitation  of  its  con- 
stituent molecules  and  atoms  as  the  temperature  is  raised,  will 
undoubtedly  produce  a  corresponding  increase  in  the  amplitude  of 
vibration  of  its  electrons  together  with  an  increase  in  their  trans- 
lational  velocity  among  the  molecules.  Under  these  conditions, 
the  specific  heat  at  constant  volume  should  be  greater  than  5.9 
calories.  This  conclusion  cannot  be  verified  experimentally  until 
the  values  of  a  and  /3  in  the  foregoing  equation  have  been  deter- 
mined at  high  temperatures. 

The  complete  heat  capacity  curves  for  three  typical  solid  ele- 
ments, lead,  aluminium,  and  carbon,  are  given  in  Fig.  49.  It  is 
apparent  from  these  curves  that  the  absorption  of  heat  energy  by 
a  crystalline  solid  may  be  considered  as  taking  place  in  three  dis- 
tinct stages,  as  follows:  —  (1)  In  the  neighborhood  of  the  abso- 
lute zero,  the  heat  capacity  remains  practically  zero;  (2)  the  heat 
capacity  increases  rapidly  with  the  temperature;  and  (3)  the  heat 
capacity  increases  slowly,  approaching  asymptotically  the  limiting 
value  5.9  for  T  =  oo.  For  a  malleable,  low  melting  element  of 
high  atomic  weight,  such  as  lead,  the  first  two  stages  are  very 
short  and  the  final  stage  commences  at  a  low  temperature.  On 
the  contrary,  with  a  hard,  high  melting  element  of  low  atomic 
weight,  such  as  carbon  in  the  form  of  diamond,  the  final  stage  is 
not  reached  at  any  temperature  within  the  range  covered  by  the 
experiments. 


164 


THEORETICAL  CHEMISTRY 


The  Nernst-Lindemann  Equation.  Recently,  several  equa- 
tions have  been  derived  expressing  the  heat  capacity  of  a  solid  in 
terms  of  temperature  and  arbitrary  constants.  All  of  these 
equations  are  based  upon  the  so-called  "  quantum  theory  "  accord- 
ing to  which  the  absorption  of  heat  energy  by  matter  is  supposed 
to  take  place  in  a  discontinuous  manner,  the  discrete  units  of 
energy  being  termed  quanta.  While  the  discussion  of  the  quantum 
theory  and  the  equations  connecting  heat  capacity  and  tempera- 
ture lies  outside  of  the  scope  of  this  book,  mention  should  never- 
theless be  made  of  the  empirical  equation  derived  by  Nernst  and 
Lindemann.*  This  equation  gives  values  of  Cv  which  are  in  re- 


100 


200  300 

T 
Fig.  49. 


400 


500 


markably  close  agreement  with  the  values  determined  by  direct 
experiment.    The  equation  may  be  written  in  the  following  form :  — 


In  this  expression,  R  is  the  molecular  gas  constant  R  =  2  calories, 
e  is  the  base  of  the  natural  system  of  logarithms  and  0  is  a  constant 
depending  upon  the  nature  of  the  solid. 

*  Zeit,  Elektrochem.,  17,  817  (1911). 


SOLIDS  165 

The  value  of  6  may  be  calculated  with  a  fair  degree  of  accuracy 
by  means  of  the  equation 


fd\i 

(A) 


in  which  T/  denotes  the  absolute  melting  point  of  the  substance, 
A  its  atomic  weight  and  d  its  density. 

Liquid  Crystals.  In  addition  to  possessing  well-defined  geo- 
metrical forms,  crystalline  substances  are  characterized  by  their 
resistance  to  deformation  when  subjected  to  mechanical  stress, 
and  by  the  property  of  melting  sharply  at  definite  temperatures 
with  the  production  of  transparent  liquids. 

In  1888,  two  substances,  cholesteryl  acetate  and  cholesteryl 
benzoate,  were  found  by  Reinitzer  *  to  behave  in  an  anomalous 
manner  when  heated.  At  definite  temperatures  these  substances 
melted  to  turbid  liquids  which,  in  turn,  became  clear  on  further 
heating,  the  latter  change  also  taking  place  at  definite  tempera- 
tures. On  cooling  the  clear  liquids,  the  reverse  series  of  changes 
was  found  to  occur. 

Examination  of  the  turbid  liquids  revealed  the  fact  that  they 
resembled  ordinary  liquids  in  their  general  behavior,  such  as  assum- 
ing the  spherical  shape  when  suspended  in  a  medium  of  the  same 
density,  or  of  rising  in  a  capillary  tube  under  the  influence  of  sur- 
face tension.  But  in  addition  to  possessing  the  properties  char- 
acteristic of  the  liquid  state,  Lehmann  discovered  that  they 
possessed  optical  properties  which  had  hitherto  been  observed 
only  with  solid,  crystalline  substances.  Their  behavior  towards 
polarized  light  was  such  as  to  warrant  the  conclusion  that  these 
turbid  liquids  are  anisotropic.  In  view  of  these  facts,  Lehmann 
proposed  that  liquids  possessing  these  properties  should  be  called 
liquid  crystals,  the  term  implying  that  under  ordinary  condi- 
tions, the  crystalline  forces  in  these  substances  are  so  feeble  that 
the  crystals  readily  undergo  deformation  and  actually  flow  like 
liquids.  That  these  turbid  liquids  are  not  emulsions,  is  proven 
by  the  fact  that  when  they  are  examined  under  the  micro- 
scope, the  turbidity  is  found  to  be  due  to  the  aggregation  of  a 
*  Monatshefte,  9,  435  (1888). 


166  THEORETICAL  CHEMISTRY 

myriad  of  differently  oriented  transparent  crystals.  All  subse- 
quent investigation  of  liquid  crystals  has  failed  to  show  any  lack 
of  homogeneity. 

The  number  of  such  substances  known  at  the  present  time  is 
fairly  large. 


CHAPTER   IX. 
SOLUTIONS. 

Classification  of  Solutions.  Having  dealt  with  the  properties 
of  pure  substances  in  the  gaseous,  liquid  and  solid  states  we  now 
proceed  to  the  consideration  of  the  properties  of  mixtures  of  two 
or  more  pure  substances.  When  such  a  mixture  is  chemically 
and  physically  homogeneous,  and  no  abrupt  change  in  its  prop- 
erties results  from  an  alteration  of  the  proportions  of  the  com- 
ponents of  the  mixture,  it  is  termed  a  solution.  When  one 
substance  is  dissolved  in  another,  it  is  customary  to  designate  as 
the  solvent  that  component  which  is  present  in  the  larger  proportion, 
the  other  component  being  termed  the  solute.  When  not  more 
than  one-tenth  mol  of  solute  is  present  in  one  liter  of  solution,  the 
solution  is  said  to  be  dilute.  The  detailed  study  of  dilute  solu- 
tions will  be  deferred  until  the  next  chapter. 

There  are  nine  possible  classes  of  solutions,  as  follows :  — 

(1)  Solution  of  gas  in  gas; 

(2)  Solution  of  liquid  in  gas; 

(3)  Solution  of  solid  in  gas; 

(4)  Solution  of  gas  in  liquid; 

(5)  Solution  of  liquid  in  liquid; 

(6)  Solution  of  solid  in  liquid; 

(7)  Solution  of  gas  in  solid; 

(8)  Solution  of  liquid  in  solid; 

(9)  Solution  of  solid  in  solid. 

While  examples  of  all  of  these  different  types  of  solutions  are^known, 
only  the  more  important  classes  will  be  considered  here. 

Solutions  of  Gases  in  Gases.  In  solutions  of  this  class  the 
components  may  be  present  in  any  proportions,  since  gases  are 
completely  miscible.  In  a  mixture  of  gases  where  no  chemical 
action  occurs,  each  gas  behaves  independently,  the  properties  of 

167 


168 


THEORETICAL  CHEMISTRY 


the  gaseous  mixture  being  the  sum  of  the  properties  of  the  con- 
stituents. Thus,  the  total  pressure  of  a  mixture  of  several  gases  is 
equal  to  the  sum  of  the  pressures  which  each  gas  would  exert  were  it 
alone  present  in  the  volume  occupied  by  the  mixture.  This  law  was 
discovered  by  Dalton  *  and  is  known  as  Dalton's  law  of  partial 
pressures.  If  the  partial  pressures  of  the  constituent  gases  be 
denoted  by  pi,  p2,  p3,  etc.,  and  P  and  V  represent  the  total  pres- 
sure and  the  total  volume  of  the  gaseous  mixture,  then 

PV   =    V  (Pl  +  P2  +  P3  +     .     .     .     )• 

Dalton's  law  holds  when  the  partial  pressures  are  not  too  great, 

its  order  of  validity  being  the  same 
as  that  of  the  other  gas  laws.  Dal- 
ton's law  can  be  tested  experimen- 
tally by  comparing  the  total  pressure 
of  the  gases  with  the  sum  of  the  pres- 
sures exerted  by  each  gas  before 
mixture.  Van't  Hoff  pointed  out 
the  possibility  of  measuring  the  par- 
tial pressure  of  one  of  the  two  com- 
ponents of  a  gas  mixture,  provided 
a  diaphragm  could  be  found  which 
would  be  pervious  to  one  of  the 
gases  but  not  to  the  other.  It  was 
shown  shortly  afterward  by  Ram- 
say,! that  the  walls  of  a  vessel  of 
palladium,  when  sufficiently  heated, 
permit  the  free  passage  of  hydrogen 
but  not  of  nitrogen.  The  walls  are 
A  sketch  of  the  apparatus  used  by 


HytLnogen 


Fig.  50. 


said  to  be  semi-permeable. 
Ramsay  in  the  verification  of  Dalton's  law  is  shown  in  Fig.  50. 
A  small  vessel  of  palladium,  P,  containing  nitrogen,  is  connected 
with  a  manometer  AB,  which  serves  to  measure  the  pressure  of  the 
gas  in  P.  The  vessel  P  is  enclosed  within  a  larger  vessel  C,  which 
can  be  filled  with  hydrogen  at  known  pressure.  On  heating  P  and 

*  Gilb.  Ann.,  12,  385  (1802). 
t  Phil.  Mag.  (5),  38,  206  (1894). 


SOLUTIONS  169 

passing  a  current  of  hydrogen  at  a  definite  pressure  through  C,  the 
hydrogen  enters  P  until  the  pressures  due  to  hydrogen  inside  and 
outside  are  equal.  The  total  pressure  in  P,  measured  on  the 
manometer,  is  greater  than  the  pressure  in  C.  The  difference 
between  the  two  pressures  is  very  nearly  equal  to  the  partial  pres- 
sure of  the  nitrogen.  Conversely,  if  a  mixture  of  the  two  gases 
be  introduced  into  P,  which  is  then  heated  and  maintained  at  suf- 
ficiently high  temperature  to  insure  its  permeability  to  hydrogen, 
the  partial  pressure  of  the  nitrogen  can  be  determined  by  passing 
a  current  of  hydrogen  at  known  pressure  through  C  until  equilib- 
rium is  attained,  as  shown  by  the  manometer.  The  difference 
between  the  external  and  internal  pressures  is  the  partial  pressure 
of  the  nitrogen.  This  experiment  has  a  very  important  bearing 
upon  the  modern  theory  of  solution. 

Solutions  of  Gases  in  Liquids.  The  solubility  of  gases  in 
liquids  is  limited,  the  extent  to  which  they  dissolve  depending 
upon  the  pressure,  the  temperature,  the  nature  of  the  gas,  and 
the  nature  of  the  solvent.  When  a  liquid  cannot  absorb  any 
more  of  a  gas  at  a  definite  temperature,  it  is  said  to  be  saturated, 
and  the  solution  is  called  a  saturated  solution.  The  solubility  of 
a  gas  in  a  liquid  is  defined  by  Ostwald  as  the  ratio  of  the  volume 
of  the  gas  absorbed  to  the  volume  of  the  absorbing  liquid  at  a 
specified  temperature  and  pressure,  or  if  the  solubility  of  the  gas 
be  represented  by  S,  we  have 

S  =  v/V, 

where  v  is  the  volume  of  gas  absorbed  and  V  is  the  volume  of  the 
absorbing  liquid.  The  " absorption  coefficient"  of  Bunsen  in 
terms  of  which  he  expressed  the  results  of  his  measurements  of 
the  solubility  of  gases,  may  be  defined  as  the  volume  of  a  gas, 
reduced  to  0°  C.  and  76  cm.  pressure  which  is  absorbed  by  unit 
volume  of  a  liquid  at  a  certain  temperature  and  under  a  pressure 
of  76  cm.  of  mercury.  In  certain  cases  the  volume  of  the  gas 
absorbed  is  found  to  be  independent  of  the  pressure,  so  that  if  a 
is  the  coefficient  of  gaseous  expansion,  and  0  Bunsen's  coefficient 
of  absorption,  then 

S  =  j8  (1  +  at). 


170 


THEORETICAL  CHEMISTRY 


The  solubilities  of  a  few  gases  in  water  and  alcohol  as  determined 
by  Bunsen  are  given  in  the  following  table :  — 


Gas. 

Water. 

Alcohol. 

0° 

15° 

0° 

15° 

Hydrogen  

0.0215 
0.0489 
1.797 

0.0190 
0.0342 
0.1002 

0.0693 
0.2337 
4.330 

0.0673 
0.2232 
3.199 

Oxygen  

Carbon  dioxide 

The  solubility  of  gases  in  water  is  appreciably  diminished  by 
the  presence  of  dissolved  solids  or  liquids,  especially  electrolytes. 
Various  theories  have  been  proposed  to  account  for  the  diminished 
solubility  of  gases  in  salt  solutions  but  the  most  satisfactory  is 
that  due  to  Philip,*  who  suggests  that  the  phenomenon  is  caused 
by  the  hydration  of  the  dissolved  salt.  A  portion  of  the  water 
in  the  salt  solution  is  supposed  to  be  in  combination  with  the  salt, 
the  water  which  is  thus  removed  from  the  role  of  solvent,  being 
no  longer  free  to  absorb  gas.  The  solubility  of  a  gas  increases 
with  increase  in  pressure.  For  gases  which  do  not  react  chem- 
ically with  the  solvent,  there  exists  a  simple  relation  between 
pressure  and  solubility,  discovered  by  Henry,  f  This  relation, 
known  as  Henry1  s  law  may  be  stated  as  follows :  —  When  a  gas  is 
absorbed  in  a  liquid,  the  weight  dissolved  is  proportional  to  the  pressure 
of  the  gas.  Since  pressure  and  volume,  at  constant  temperature, 
are  inversely  proportional  (Boyle's  law),  the  law  of  Henry  may  be 
stated  thus :  —  The  volume  of  a  gas  absorbed  by  a  given  volume  of 
liquid  is  independent  of  the  pressure.  There  is  yet  another  form 
in  which  the  law  may  be  stated  which  is  instructive  in  connection 
with  the  modern  theory  of  solution.  When  a  definite  volume  of 
liquid  is  saturated  with  a  gas  at  constant  temperature  and  pres- 
sure, a  condition  of  equilibrium  is  established  between  the  gas  in 
solution  and  that  in  the  free  space  over  the  solution,  therefore, 
Henry's  law  may  be  stated  as  follows: —  The  concentration  of  the 
dissolved  gas  is  directly  proportional  to  that  in  the  free  space  above 

*  Trans.  Faraday  Soc.,  3,  140  (1907). 
f  Gilb.  Ann.,  20,  147  (1805). 


SOLUTIONS  171 

the  liquid.  If  c\  represents  the  concentration  of  the  gas  in  the 
liquid  and  Cz  the  concentration  in  the  free  space  above  the  liquid, 
Henry's  law  may  be  expressed  thus :  — 

ci/cz  =  k, 

where  k  is  known  as  the  solubility  coefficient. 

Dalton  showed  that  the  solubility  of  the  individual  gases  in  a 
mixture  of  gases  is  directly  proportional  to  their  partial  pressures, 
the  solubility  of  each  gas  being  nearly  independent  of  the  presence 
of  the  others. 

As  will  be  seen  from  the  foregoing  table,  the  solubility  of  a  gas 
in  a  liquid  diminishes  with  increase  in  temperature.  Concerning 
the  influence  of  the  nature  of  the  gas  on  its  solubility,  it  may  be 
said  that  those  gases  which  exhibit  acid  or  basic  reactions  are 
the  most  soluble,  the  solubilities  of  neutral  gases  being  small. 
In  the  case  of  many  of  the  very  soluble  gases  Henry's  law  does 
not  hold.  For  example,  ammonia,  a  gas  having  marked  basic 
properties  and  a  large  coefficient  of  solubility,  does  not  obey 
Henry's  law  at  ordinary  temperatures,  the  mass  of  ammonia 
absorbed  not  being  proportional  to  the  pressure.  The  curve 
showing  the  variation  in  solubility  with  pressure  at  0°  C.  has 
two  marked  discontinuities.  At  temperatures  above  100°  C. 
the  gas  obeys  Henry's  law.  Sulphur  dioxide  behaves  similarly, 
the  law  holding  only  for  temperatures  exceeding  40°  C. 

With  regard  to  the  connection  between  the  solvent  power  of  a 
liquid  and  its  nature  but  little  is  known.  About  all  that  can  be 
said  is,  that  the  order  of  solubility  of  gases  in  different  liquids  is 
the  same.  Thus  in  the  preceding  table  the  solubilities  of  hydro- 
gen, oxygen  and  carbon  dioxide  in  water  and  in  alcohol  will  be 
seen  to  be  approximately  proportional.  A  slight  change  in 
volume  always  results  when  a  gas  is  dissolved  in  a  liquid.  In 
general  it  may  be  said  that  the  less  compressible  a  gas  is,  the 
greater  is  the  increase  in  volume  produced  when  it  is  absorbed  by 
a  liquid.  It  is  of  interest  to  note  that  the  increase  in  volume 
caused  by  the  solution  of  a  gas  is  nearly  equal  to  the  value  of  b 
for  the  gas  in  the  equation  of  van  der  Waals.  This  is  shown  in 
the  following  table :  — 


172  THEORETICAL  CHEMISTRY 


Gas. 

Increase  in  Vol. 

6 

Oxygen 

0.00115 
0.00145 
0.00106 
0.00125 

0.000890 
0.001359 
0.000887 
0.000866 

Nitrogen 

Hydrogen  . 
Carbon  dio 

xide                              

Solutions  of  Liquids  in  Liquids.  Solutions  of  liquids  in  liquids 
can  be  divided  into  three  classes  as  follows:  —  (1)  Liquids  which 
are  miscible  in  all  proportions;  (2)  Liquids  which  are  partially 
miscible;  and  (3)  Liquids  which  are  immiscible.  Examples  of 
these  three  classes  in  the  order  mentioned  are,  alcohol  and  water, 
ether  and  water,  and  benzene  and  water.  As  to  the  cause  of 
miscibility  and  non-miscibility  of  liquids  very  little  is  known. 

Partial  Miscibility.  If  a  small  amount  of  ether  is  added  to 
a  large  volume  of  water  in  a  separatory  funnel  and  the  mixture 
vigorously  shaken,  a  perfectly  homogeneous  solution  will  be 
obtained.  On  gradually  increasing  the  amount  of  ether,  shaking 
after  each  addition,  a  concentration  will  eventually  be  reached 
at  which  a  separation  into  two  layers  will  take  place.  The  upper 
layer  is  a  saturated  solution  of  water  in  ether  and  the  lower  layer 
is  a  saturated  solution  of  ether  in  water.  So  long  as  the  relative 
amounts  of  the  two  liquids  is  such  that  the  mixture  does  not 
become  homogeneous  on  standing,  the  composition  of  the  two 
layers  will  be  independent  of  the  relative  amounts  of  the  two 
components.  Measurements  of  the  mutual  solubility  of  liquids 
have  been  made  by  Alexieeff  *  by  placing  weighed  amounts  in 
sealed  tubes  and  observing  the  temperature  at  which  the  mixture 
became  homogeneous.  In  general  the  solubility  of  a  pair  of 
partially  miscible  liquids  increases  with  the  temperature,  and 
therefore  it  may  be  inferred  that  at  a  sufficiently  high  temperature 
the  mixture  will  become  perfectly  homogeneous.  An  example  of 
this  type  of  binary  mixture  is  furnished  by  phenol  and  water,  the 
solubility  curve  of  which  is  shown  in  Fig.  51.  In  this  diagram 
temperature  is  plotted  on  the  axis  of  ordinates  and  percentage 
composition  of  the  solution  on  the  axis  of  abscissae.  Starting 

*  Jour,  prakt.  Chem.,  133,  518  (1882);  Bull.  Soc.  Chem.,  38,  145  (1882). 


SOLUTIONS 


173 


with  a  small  amount  of  phenol  and  adding  it  in  increasing  quan- 
tities to  a  large  volume  of  water,  a  concentration  will  eventually 
be  reached  at  which  the  solution  will  separate  into  two  layers. 
This  concentration  is  represented  by  the  point  A.  On  raising  the 


100  # 


Percentage  Water  in  Phenol 


Percentage  Phenol  in  Water 
Fig.  51. 


100  * 


temperature,  the  solubility  of  phenol  in  water  increases,  as  shown 
by  the  curve  AB.  In  like  manner,  starting  with  pure  phenol 
and  adding  increasing  amounts  of  water,  separation  into  two  layers 
will  occur  at  a  concentration  represented  by  the  point  C.  As  the 
temperature  is  raised  the  solubility  of  water  in  phenol  increases, 
as  shown  by  the  curve  CB.  When  the  temperature  is  raised 
above  68°.4  C.,  corresponding  to  the  point  B,  phenol  and  water 
become  miscible  in  all  proportions. 

If  we  start  with  a  solution  whose  temperature  and  composition 
is  represented  by  the  point  a,  the  addition  of  increasing  amounts 
of  phenol,  at  constant  temperature  will  be  represented  by  the 
dotted  line  afed.  When  the  point  /  is  reached,  the  solution  will 
separate  into  two  layers  the  composition  of  which  will  be  inde- 
pendent of  the  relative  amounts  of  phenol  and  water.  At  e  the 
solution  will  again  become  homogeneous.  If  the  solution  repre- 


174  THEORETICAL   CHEMISTRY 

sented  by  the  point  a  be  again  chosen  as  the  starting  point,  and  its 
composition  be  kept  unaltered  while  the  temperature  is  raised  to  a 
value  above  68°.4  C.,  the  change  will  be  represented  by  the  dotted 
line  ab.  If  now  the  temperature  be  maintained  constant  and  the 
percentage  of  phenol  increased,  the  alteration  in  composition  will 
be  effected  without  discontinuity,  as  represented  by  the  dotted 
line  be.  On  cooling  the  solution  represented  by  the  point  c  to  the 
initial  temperature  of  a,  the  point  d  will  be  reached.  Thus  it  is 
possible  to  pass  from  a  to  d  by  the  path  abed  without  causing  a 
separation  of  the  components  into  two  layers.  There  is  an 
analogy  between  the  solubility  curve  of  a  pair  of  partially  mis- 
cible  liquids  and  the  dotted,  parabolic  curve  in  the  diagram  of 
the  isothermals  of  carbon  dioxide,  shown  in  Fig.  23.  In  both 
cases  there  is  but  one  phase  outside  of  the  curves,  while  two 
phases  are  coexistent  within  the  area  enclosed  by  the  curves.  The 
analogy  may  be  traced  further,  since  in  each  case  only  one  phase 
can  exist  above  a  certain  temperature.  The  temperature  corre- 
sponding to  the  apex  of  the  parabolic  curve  in  Fig.  23,  is  termed 
the  critical  temperature  of  carbon  dioxide  and  by  analogy  the 
temperature  corresponding  to  the  point,  B,  in  Fig.  51  is  called  the 
critical  solution  temperature.  The  mutual  solubilities  of  some 
pairs  of  partially  miscible  liquids  were  found  by  Alexieeff  to  di- 
minish with  increasing  temperature.  Thus  a  mixture  of  ether  and 
water,  which  is  perfectly  homogeneous  at  ordinary  temperatures, 
becomes  turbid  on  warming.  A  specially  interesting  pair  of 
liquids  is  nicotine  and  water.  At  ordinary  temperatures  these 
liquids  are  miscible  in  all  proportions.  If  the  temperature  is 
raised  above  60°  C.,  the  solution  becomes  turbid  owing  to  incom- 
plete miscibility.  On  continuing  to  heat  the  mixture  the 
mutual  solubility  of  the  liquids  begins  to  increase,  until  at 
210°  C.  they  become  completely  soluble  again.  The  solubility 
relations  of  this  binary  mixture  are  shown  in  Fig.  52.  The  closed 
solubility  curve  defines  the  limits  of  the  coexistence  of  two  layers, 
all  points  outside  of  the  curve  representing  homogeneous  solu- 
tions. 

Complete  Miscibility.     The  study  of  the  vapor  pressures   of 
binary  mixtures  of  completely  miscible  liquids  is  of  great  im- 


SOLUTIONS 


175 


portance  in  connection  with  the  possibility  of  separating  them 
by  the  process  of  distillation.  The  experimental  investigations  of 
Konowalow  *  on  homogeneous  binary  mixtures  of  liquids  have 
shown  that  such  pairs  of  liquids  may  be  divided  into  three  classes 


100  i 


Percentage  Water  in.Nicotine 


210° 


0  #  Percentage  Nicotine  in  Water 

Fig.  52. 


100* 


as  follows: — (1)  Mixtures  having  a  maximum  vapor  pressure 
corresponding  to  a  certain  composition,  e.g.,  propyl  alcohol  and 
water;  (2)  Mixtures  having  a  minimum  vapor  pressure  corre- 
sponding to  a  certain  composition,  e.g.,  formic  acid  and  water; 
and  (3)  Mixtures  having  vapor  pressures  intermediate  between 
the  vapor  pressures  of  the  pure  components,  e.g.,  methyl  alcohol 
and  water.  In  considering  the  possibility  of  separating  binary 
mixtures  of  liquids  belonging  to  these  three  classes,  it  is  essential 
to  determine  the  composition  of  both  solution  and  escaping  vapor. 
When  a  pure  liquid  is  boiled  the  composition  of  the  escaping 
vapor  is  the  same  as  that  of  the  liquid  itself,  but  this  is,  in  general, 

*  Wied.  Ann.,  14,  34  (1881). 


176 


THEORETICAL  CHEMISTRY 


not  the  case  when  a  binary  mixture  is  distilled.  The  composition 
of  the  liquid  mixture  in  the  distilling  flask  generally  alters  contin- 
uously when  such  a  mixture  is  distilled. 

(1)  The  relation  between  the  vapor  pressure  and  composition 
of  all  possible  mixtures  of  propyl  alcohol  and  water  is  represented 
graphically  in  Fig.  53.  In  this  diagram  the  compositions  of  the 
mixtures  are  plotted  as  abscissae  and  vapor  pressures  as  ordinates. 
The  vapor  pressures  of  the  pure  components,  water  and  propyl 
alcohol,  at  a  definite  temperature  are  represented  by  A  and  C. 
The  maximum  in  the  vapor-pressure  curve  corresponds  to  a  mix- 


100* 


Water 


Propyl  Alcohol         8Q#          100* 
Fig.  53. 

ture  containing  80  per  cent  of  propyl  alcohol.  The  dotted  curve 
represents  the  boiling-points  of  the  various  mixtures  under  normal 
atmospheric  pressure.  Konowalow  has  shown  that  the  vapor  of 
a  binary  mixture  with  a  minimum  or  maximum  boiling-point  has 
the  same  composition  as  that  of  the  liquid.  The  vapor  of  all 
mixtures  containing  less  than  80  per  cent  of  propyl  alcohol  will 
be  relatively  richer  in  alcohol  than  the  liquid  mixture,  since  the 
vapor  of  propyl  alcohol  is  quite  insoluble  in  water.  If  the  amount 
of  alcohol  in  the  mixture  exceeds  80  per  cent,  then  the  vapor  will 
be  relatively  richer  in  water.  Thus,  whatever  may  be  the  com- 
position of  the  mixture  in  the  distilling  flask,  the  distillate  will 
approximate  to  the  composition  of  the  mixture  having  the  mini- 
mum boiling-point.  The  residue  in  the  flask  will  gradually 


SOLUTIONS 


,177 


change  to  pure  water  if  the  original  concentration  were  below 
80  per  cent,  or  to  pure  alcohol  if  the  original  concentration  were 
above  80  per  cent. 

(2)  The  second  type  of  binary  mixture  of  liquids  is  illustrated 
by  formic  acid  and  water,  the  vapor  pressure  and  boiling-point 
curves  for  which  are  shown  in  Fig.  54,  A  mixture  containing 
73  per  cent  of  formic  acid  has  a  minimum  vapor  pressure  and  a 
maximum  boiling-point.  At  this  concentration  the  vapor  and  the 
liquid  have  the  same  composition.  The  vapor  of  mixtures  con- 


oo  f 


Water 


760  totofl 


Porffiic  Acid- 
Fig.  54. 


100 1 


taining  less  than  73  per  cent  of  acid  is  relatively  richer  in  water 
than  the  liquid,  while  the  vapor  of  mixtures  containing  more  than 
73  per  cent  of  acid  contains  relatively  less  water  than  the  liquid. 
Any  mixture  of  formic  acid  and  water  when  distilled  will  thus 
leave  a  residue  in  the  distilling  flask  containing  73  per  cent  of 
acid;  this  residue  will  distil  at  constant  temperature  like  a  homo- 
geneous liquid.  It  was  thought  for  a  long  time  that  such  constant 
boiling  mixtures  were  definite  chemical  compounds  of  the  two 
liquids.  Thus  a  mixture  of  hydrochloric  acid  and  water  contain- 
ing 20.2  per  cent  of  acid  boils  at  110°  C.  under  atmospheric  pres- 
sure. The  composition  of  such  a  mixture  corresponds  very  nearly 
to  the  formula,  HC1.8  H20.  Roscoe  *  showed  that  these  mix- 

*  Lieb.  Ann.  116,  203  (1860). 


178 


THEORETICAL  CHEMISTRY 


tures  are  not  definite  chemical  compounds  since  the  composition 
of  the  distillate  changes  when  the  distillation  is  carried  out  under 
different  pressures. 

(3)  The  vapor-pressure  and  boiling-point  curves  for  methyl 
alcohol  and  water,  a  mixture  typical  of  the  third  class  of  com- 
pletely miscible  liquids,  are  shown  in  Fig.  55,  the  heavy  line 
representing  vapor  pressures  at  65°.2  C.  and  the  dotted  line  the 
boiling  points  under  normal  atmospheric  pressure.  In  this  case 


100  # 


Water 


Methyl  Alcohol 
Fig.  55. 


100* 


the  composition  of  both  vapor  and  liquid  alter  continuously  on 
distillation.  The  .distillate  will  contain  a  relatively  larger  amount 
of  alcohol  and  the  residue  in  the  distilling  flask,  an  excess  of  water. 
If  this  distillate  be  redistilled  from  a  clean  flask,  a  second  dis- 
tillate still  richer  in  alcohol  will  be  obtained.  By  repeating  this 
process  a  sufficient  number  of  times,  a  more  or  less  complete 
separation  of  the  two  components  of  the  mixture  can  be  effected. 
This  process  is  termed  fractional  distillation. 

Immistibility.  When  two  immiscible  liquids  are  brought 
together,  the  total  vapor  pressure  is  equal  to  the  sum  of  the  vapor 
pressures  of  the  components;  hence  when  such  a  mixture  is  dis- 
tilled, the  two  liquids  will  pass  over  in  the  ratio  of  their  respective 


SOLUTIONS  179 

vapor  pressures,  the  boiling-point  of  the  mixture  being  the  temper- 
ature at  which  the  sum  of  the  vapor  pressures  of  the  two  liquids 
is  equal  to  the  pressure  of  the  atmosphere.  The  relation  between 
vapor  pressure  and  composition  in  this  case  will  be  represented 
by  a  horizontal  line  drawn  at  a  distance  above  the  axis  of  abscissae 
equal  to  the  sum  of  the  vapor  pressures  of  the  components. 

Nitrobenzene  and  water  may  be  chosen  as  an  example  of  a  pair 
of  liquids  which  are  practically  immiscible.  Under  a  pressure 
of  760  mm.  the  mixture  boils  at  99°  C.  The  vapor  pressure  of 
water  at  this  temperature  is  733  mm.  ;  the  vapor  pressure  of  nitro- 
benzene must  be  760  —  733  =  27  mm.  Notwithstanding  the 
relatively  small  vapor  pressure  of  nitrobenzene  in  the  mixture, 
considerable  quantities  of  it  distil  over  with  the  water.  It  is  this 
fact  that  makes  possible  separations  of  liquids  by  the  process  of 
steam  distillation  so  frequently  employed  by  the  organic  chem- 
ist. The  relative  weights  of  water  and  nitrobenzene  passing 
over  in  a  steam  distillation  may  be  calculated  as  follows  :  —  The 
relative  volumes  of  steam  and  vapor  of  nitrobenzene  which  distil 
over  will  be  in  the  ratio  of  their  respective  vapor  pressures  at  the 
temperature  of  the  experiment,  and  consequently  the  relative 
weights  of  the  two  liquids  which  pass  over  will  be  in  the  ratio, 
Pidi  :  pzdz,  where  pi  and  p2  denote  the  respective  vapor  pressures 
of  water  and  nitrobenzene,  and  di  and  d2  the  corresponding  vapor 
densities.  If  Wi  and  w2  denote  the  weights  of  the  two  liquids  in 
the  state  of  vapor,  then 


wi  :  w2  ::  pii 

or,  since  vapor  density  is  proportional  to  molecular  weight,  we 
may  write 

w  ii  w2  ::  piMi  :  p2M2. 

Substituting  in  this  proportion  the  values  given  above  for  the  vapor 
pressures  of  steam  and  nitrobenzene,  we  have 

WL  :  w2  ::  733  X  18  :  27  X  123 
or, 

it*  :«*::  13,194:  3321. 

Thus  the  weights  of  water  and  nitrobenzene  in  the  distillate  are 
approximately  in  the  ratio  of  4  to  1  notwithstanding  the  fact  that 


180 


THEORETICAL  CHEMISTRY 


the  ratio  of  their  vapor  pressures  at  the  boiling-point  of  the  mix- 
ture is  27  to  1.  If  an  organic  substance  is  not  decomposed  by 
steam,  it  is  possible  to  effect  an  appreciable  purification  by  steam 
distillation,  even  though  its  vapor  pressure  be  relatively  small. 
As  will  be  seen  from  the  above  example,  it  is  the  high  molecular 
weight  of  the  nitrobenzene  which  compensates  for  its  low  vapor 
pressure.  It  is  the  small  molecular  weight  of  water  which  renders 
it  so  suitable  for  steam  distillation. 

Finally,  the  vapor-pressure  and  boiling-point  relations  of  binary 
mixtures  of  partially  miscible  liquids  must  be  considered.     In 


100  # 


Water 


/B 

c\ 

'A 

\/ 

• 

\ 

X 

1  \ 

\ 
\ 

/  D 

\ 

\ 

\ 

/ 

\ 

\ 

760  mm.-                            / 

B7" 

c 

Of 

Isobutyl  Alcohol                           10( 
Fig.  56. 

general  when  two  liquids  are  mixed,  each  lowers  the  vapor  pressure 
of  the  other,  so  that  the  vapor  pressure  of  the  mixture  is  less  than 
the  sum  of  the  vapor  pressures  of  the  components.  As  has  al- 
ready been  pointed  out,  the  composition  of  the  two  layers  in  a 
binary  mixture  of  partially  miscible  liquids  is  independent  of 
the  relative  amounts  of  the  components  present;  hence  the  vapor 
pressure  remains  constant  so  long  as  the  solution  remains  hetero- 
geneous. The  vapor-pressure  and  boiling-point  curves  for  a 
binary  mixture  of  partially  miscible  liquids,  (isobutyl  alcohol  and 
water),  are  shown  in  Fig.  56.  The  horizontal  portion  BC,  repre- 
sents the  vapor  pressures,  at  88°.5  C.,  of  mixtures  of  isobutyl 
alcohol  and  water  where  two  layers  are  present.  -The  vapor 


SOLUTIONS  181 

pressure  of  the  homogeneous  mixtures  are  represented  by  AB  and 
CD,  AB  corresponding  to  solutions  of  isobutyl  alcohol  in  water, 
and  CD  to  solutions  of  water  in  isobutyl  alcohol.  The  dotted 
line  A'B'C'D'  represents  the  boiling-points  of  all  possible  mix- 
tures of  isobutyl  alcohol  and  water,  under  normal  atmospheric 
pressure. 

Solutions  of  Solids  in  Liquids.  The  solubility  of  a  solid  in  a 
liquid  is  limited  and  is  dependent  upon  the  temperature,  the 
nature  of  the  solute  and  the  nature  of  the  solvent.  When  a 
solvent  has  taken  up  as  much  of  a  solute  as  it  is  capable  of  dis- 
solving at  a  definite  temperature,  the  solution  is  said  to  be  satu- 
rated. There  are  two  general  methods  for  the  preparation  of 
saturated  solutions: —  (1)  An  excess  of  the  finely-divided  solute 
is  agitated  with  a  known  amount  of  the  solvent,  at  a  definite 
temperature,  until  equilibrium  is  attained;  (2)  the  solvent  is" 
heated  with  an  excess  of  the  solute  to  a  temperature  higher  than 
that  at  which  saturation  is  required,  and  then  cooled  in  contact 
with  the  solid  solute  to  the  desired  temperature.  Both  of  these 
methods  give  equally  satisfactory  results  provided  sufficient 
time  is  allowed  for  the  establishment  of  equilibrium,  and  provided 
the  solid  substance  is  always  present  in  excess.  The  solubility 
of  a  solid  in  a  liquid  may  be  expressed  as  the  number  of  grams  of 
the  solute  in  a  given  mass  or  volume  of  solvent  or  solution,  but 
it  is  usually  expressed  as  the  number  of  grams  of  solute  in  100 
grams  of  solution.  .  The  solubility  of  solids  has  recently  been 
shown  to  be  somewhat  dependent  upon  their  state  of  division. 
Thus,  Hulett  *  has  found  that  a  saturated  solution  of  gypsum  at 
25°  C.  contains  2.080  grams  of  CaS04  per  liter,  whereas  when 
very  finely  divided  gypsum  is  shaken  with  this  solution,  it  is 
possible  to  increase  the  content  of  dissolved  CaSCX  to  2.542  grams 
per  liter.  When  a  saturated  solution  is  cooled,  every  trace  of 
solid  solute  being  excluded,  the  excess  of  dissolved  solid  may  not 
separate.  Such  a  solution  is  said  to  be  supersaturated. 

As  a  general  rule  the  solubility  of  solids  in  liquids  increases 
with  the  temperature,  as  shown  in  Fig.  57.      Several  exceptions 
to  this  rule  are  known,  among  which  may  be  mentioned  calcium 
*  Jour.  Am.  Chem.  Soc.,  27,  49  (1905). 


182 


THEORETICAL  CHEMISTRY 


hydroxide,  calcium  sulphate  above  40°  C.,  and   sodium   sulphate 
between  the  temperatures  of  33°  C.  and  100°  C. 

Solubility  curves  are  usually  continuous,  but  exceptions  to  this 
rule  are  common:   the  solubility  curve  of  sodium  sulphate  fur- 


100 


40  60 

Temperature 

Fig.  57. 


nishes  an  illustration.  The  discontinuity  in  the  solubility  curve 
of  sodium  sulphate  is  due  to  the  fact  that  we  are  not  dealing  with 
one  solubility  curve,  but  with  two  solubility  curves.  At  temper- 
atures below  33°  C.,  the  dissolved  salt  is  in  equilibrium  with  the 
decahydrate,  Na2S04.10  H20,  whereas  at  temperatures  above 
33°  C.  the  dissolved  salt  is  in  equilibrium  with  the  anhydrous 
salt,  Na2S04.  The  solubility  of  Na2S04.10H2O  increases  with 
the  temperature,  while  the  solubility  of  Na2S04  diminishes.  That 
we  are  actually  dealing  with  two  solubility  curves,  is  proved  by 
the  fact  that  the  solubility  curves  of  the  hydrated  and  anhydrous 
salts  in  supersaturated  solutions  are  continuations  of  the  corre- 
sponding curves  for  saturated  solutions,  as  shown  by  the  dotted 


SOLUTIONS  183 

curves  in  Fig.  57.  If  we  select  any  point,  such  as  p,  lying  between 
a  dotted  curve  and  a  full  curve,  it  is  apparent  that  it  represents  a 
solution  supersaturated  with  respect  to  Na2SO4.10  H2O,  but  un- 
saturated  with  respect  to  NaaSO^  If  pure  anhydrous  sodium 
sulphate  be  shaken  with  this  solution  it  will  slowly  dissolve,  where- 
as if  a  trace  of  the  hydrated  salt  be  added,  the  solution  will  deposit 
NasSC^.lO  H20,  until  the  amount  remaining  in  solution  corre- 
sponds to  the  solubility  of  the  hydrate  at  that  temperature. 
Supersaturated  solutions  of  some  substances  can  be  preserved 
indefinitely,  provided  all  traces  of  the  solid  phase  are  excluded. 
Such  solutions  are  called  metastable.  On  the  other  hand  there 
are  some  supersaturated  solutions  which  deposit  the  excess  of 
solid  solute  even  when  all  traces  of  it  are  excluded.  These  solu- 
tions are  termed  labile.  The  distinction  between  metastable  and 
labile  solutions  is  not  sharp.  If  a  metastable  solution  is  suffi- 
ciently cooled,  or  if  its  concentration  is  sufficiently  increased,  it 
may  be  made  to  pass  over  into  the  labile  condition.  The  concen- 
tration at  which  this  transition  occurs  is  termed  the  metastable 
limit.  The  stability  of  supersaturated  solutions  has  recently 
been  shown  by  Young  *  to  be  greatly  influenced  by  vibrations  or 
sudden  shocks  within  the  solution.  He  has  been  able  to  control 
the  amount  of  overcooling  in  a  supersaturated  solution,  by  alter- 
ing the  intensity  of  the  vibrations  due  to  the  friction  between  glass 
or  metal  surfaces  within  the  solution. 

Very  little  is  known  concerning  the  relation  between  solubility 
and  the  specific  properties  of  solute  and  solvent. 

Owing  to  the  fact  that  the  change  in  volume  resulting  from  the 
solution  of  a  solid  in  a  liquid  is  very  small,  the  effect  of  pressure 
on  the  solution  is  almost  negligible.  The  chief  factors  condition- 
ing the  change  in  solubility  due  to  increasing  pressure,  are  the 
heat  of  solution  of  the  solute  in  the  nearly  saturated  solution, 
and  the  change  in  volume  on  solidification.  Very  few  experiments 
have  been  made  to  determine  the  effect  of  pressure  on  solubility. 
Van't  Hoff  states  that  the  solubility  of  a  solution  of  ammonium 
chloride,  a  salt  which  expands  when  dissolved,  decreases  by  1  per 
cent  for  160  atmospheres,  while  the  solubility  of  copper  sulphate, 

*  Jour.  Am.  Chem.  Soc.,  33,  148  (1911). 


184 


THEORETICAL  CHEMISTRY 


a  salt  which  contracts  when  dissolved,  increases  by  3.2  per  cent 
for  60  atmospheres. 

Solid  Solutions.  In  general,  when  a  dilute  solution  is  suffi- 
ciently cooled  the  solvent  separates  in  the  form  of  crystals  which 
are  almost  entirely  free  from  the  solute.  When,  however,  the 
temperature  of  a  solution  of  iodine  in  benzene  is  reduced  to  the 
freezing-point,  the  crystals  which  separate  are  found  to  contain 
iodine.  Furthermore,  the  depression  of  the  freezing-point  of  the 
solvent  is  found  to  be  less  than  that  calculated  on  the  assumption 
that  the  solvent  crystallizes  uncontaminated  with  the  solute. 
Such  solutions  were  first  studied  by  van't  Hoff.*  He  found  that 
when  the  concentration  of  such  abnormal  solutions  is  varied,  the 
ratio  of  the  amount  of  solute  in  the  liquid  solvent  to  the  amount 
of  solute  in  the  solidified  solvent  remains  constant.  Thus,  in  solu- 
tions of  iodine  in  benzene,  the  ratio  of  the  concentration  of  iodine 
in  the  liquid  to  its  concentration  in  the  crystallized  benzene  is 
constant.  In  the  following  table  Ci  is  the  concentration  of  iodine 
in  the  liquid  benzene,  and  Cz  is  the  concentration  of  iodine  in  the 
solid  benzene. 


Cl 

C2 

Cl/Cj 

3.39 

2.587 
0.945 

1.279 
0.925 
0.317 

0.377 
0.358 
0.336 

Van't  Hoff  pointed  out  the  analogy  between  the  distribution  of 
the  solute  between  the  solid  and  liquid  solvent,  and  the  distribution 
of  a  gas  between  a  liquid  and  the  free  space  above  it.  In  other 
words,  the  distribution  follows  Henry's  law  for  the  solution  of  a 
gas  in  a  liquid.  Shice  the  crystals  containing  both  solute  and 
solvent  are  perfectly  homogeneous,  van't  Hoff  suggested  that 
they  be  regarded  as  solid  solutions.  The  mixed  crystals  which 
separate  from  solutions  of  isomorphous  substances  being  chem- 
ically and  physically  homogeneous,  are  to  be  considered  as  solid 
solutions.  Many  alloys  possess  the  properties  characteristic  of 

*  Zeit.  phys.  Chem.,  5,  322  (1890). 


SOLUTIONS  185 

solid  solutions;  hardened  steel,  for  example,  being  regarded  as  a 
homogeneous  solid  solution  of  carbon  in  iron.  One  of  the  char- 
acteristic properties  of  a  dissolved  substance  is  its  tendency  to 
diffuse  into  the  pure  solvent.  Interesting  experiments  performed 
by  Roberts- Austen  *  have  shown  that  even  solids  have  the  prop- 
erty of  mixing  by  diffusion.  Thus,  by  keeping  gold  and  lead  in 
contact  at  constant  temperature  for  four  years,  he  was  able  to 
detect  the  presence  of  gold  in  the  layer  of  lead  at  a  distance  of 
7  mm.  from  the  surface  of  separation.  Many  other  instances  of 
diffusion  in  solids  have  been  observed. f 

Instances  of  gases  and  liquids  dissolving  in  solids  are  also 
known.  Thus  platinum,  palladium,  charcoal  and  other  sub- 
stances have  the  property  of  taking  up  large  volumes  of  hydrogen. 
This  phenomenon,  known  as  occlusion,  is  but  little  understood. 
Van't  Hoff  has  suggested  that  when  hydrogen  dissolves  in  palla- 
dium we  are  really  dealing  with  two  solid  solutions:  one  a  solu- 
tion of  hydrogen  in  palladium  and  the  other  a  solution  of 
palladium  in  solid  hydrogen,  the  system  being  analogous  to 
that  of  two  partially  miscible  liquids. 

Certain  natural  silicates,  the  so-called  zeolites,  are  transparent 
and  homogeneous.  Since  they  contain  varying  quantities  of 
water  they  may  be  regarded  as  examples  of  solutions  of  liquids 
in  solids.  This  classification  is  further  justified  by  the  fact  that 
portions  of  the  water  may  be  removed  and  replaced  by  other 
substances,  such  as  alcohol,  with  apparently  no  change  in  the 
transparency  or  homogeneity  of  the  mineral. 

PROBLEMS. 

1.  2.3  liters  of  hydrogen  under  a  pressure  of  78  cm.  of  mercury,  and 
5.4  liters  of  nitrogen  at  a  pressure  of  46  cm.  were  introduced  into  a  vessel 
containing  3.8  liters  of  carbon  dioxide  under  a  pressure  of  27  cm.     What 
was  the  pressure  of  the  mixture?  Ans.    140  cm.  of  mercury. 

2.  Air  is  composed  of  20.9  volumes  of  oxygen  and  79.1  volumes  of 
nitrogen.    At  15°  C.  water  absorbs  0.0299  volumes  of  oxygen  and  0.0148 

*  Proc.  Roy.  Soc.,  67,  101  (1900). 

t  See  Report  on  Diffusion  in  Solids,  by  C.  H.  Desch,  Chem.  Newg,  106, 
153  (1912). 


186  THEORETICAL  CHEMISTRY 

volumes  of  nitrogen,  the  pressure  of  each  being  that  of  the  atmosphere. 

Calculate  the  composition  of  the  mixture  of  gases  absorbed  by  the  water. 

Ans.  34.8%  by  vol.  of  O,  and  65.2%  by  vol.  of  N. 

3.  The  vapor  pressure  of  the  immiscible  liquid  system,  aniline-water, 
is  760  mm.  at  98°  C.    The  vapor  pressure  of  water  at  that  temperature  is 
707  mm.    What  fraction  of  the  total  weight  of  the  distillate  is  aniline. 

Ans.  0.28. 

4.  The  boiling-point  of  the  immiscible  liquid  system,  naphthalene-water, 
is  98°  C.  under  a  pressure  of  733  mm.    The  vapor  pressure  of  water  at 
98°  C.  is  707  mm.    Calculate  the  proportion  of  naphthalene  in  the  dis- 
tillate. Ans.  0.207. 


CHAPTER   X. 
DILUTE   SOLUTIONS   AND   OSMOTIC   PRESSURE. 

Osmotic  Pressure.  In  the  preceding  chapter  reference  was 
made  to  the  fact  that  diffusion  is  a  characteristic  property  of  solu- 
tions. If  a  few  cubic  centimeters  of  a  concentrated  solution  of 
cane  sugar  are  placed  at  the  bottom  of  a  tall  cylinder,  and  water 
is  added,  care  being  taken  to  prevent  mixture,  the  sugar  immedi- 
ately begins  to  diffuse  into  the  water,  the  process  continuing  until 
the  concentration  of  sugar  is  the  same  throughout  the  liquid. 
The  sugar  molecules  move  from  a  region  of  high  concentration 
to  a  region  of  low  concentration,  the  rate  of  diffusion  being  rela- 
tively slow  owing  to  the  viscosity  of  the  medium.  A  similar 
process  is  encountered  in  the  study  of  gases,  but  the  rate  of  gas- 
eous diffusion  is  extremely  rapid.  In  terms  of  the  kinetic  theory, 
the  movement  of  the  molecules  of  a  gas  from  regions  of  high 
concentration  to  regions  of  low  concentration,  is  to  be  considered 
as  due  to  the  pressure  of  the  gas.  By  analogy,  we  may  regard  the 
process  of  diffusion  in  solutions  as  a  manifestation  of  a  driving 
force,  known  as  the  osmotic  pressure. 

Semi-permeable  Membranes.  The  use  of  a  semi-permeable 
membrane  for  the  measurement  of  the  partial  pressure  of  nitrogen 
in  a  mixture  of  nitrogen  and  hydrogen,  has  already  been  explained. 
A  similar  method  may  be  employed  for  the  measurement  of 
the  osmotic  pressure  of  a  solution,  provided  a  suitable  semi-perme- 
able membrane  can  be  found.  Such  a  membrane  must  prevent 
the  passage  of  the  molecules  of  solute  and  must  be  readily  perme- 
able to  the  molecules  of  solvent;  it  must  exert  a  selective  action 
on  solute  and  solvent.  If  a  solution  is  separated  from  the  pure 
solvent  by  a  semi-permeable  membrane,  diffusion  of  the  solute 
is  no  longer  possible.  Since  equilibrium  of  the  system  can  only 
be  attained  when  the  concentrations  on  both  sides  of  the  mem- 
brane are  equal,  it  follows  that  the  solvent  must  pass  through 

187 


188  THEORETICAL  CHEMISTRY 

the  membrane  and  dilute  the  more  concentrated  solution.  A 
number  of  semi-permeable  membranes  have  been  discovered 
which  are  readily  permeable  to  water  and  nearly,  if  not  entirely, 
impermeable  to  various  solutes.  About  the  middle  of  the  eight- 
eenth century  Abbe  Nollet  discovered  that  certain  animal  mem- 
branes are  permeable  to  water  but  not  to  alcohol. 

Artificial  semi-permeable  membranes  were  first  prepared  by 
M.  Traube.*  If  a  glass  tube,  provided  with  a  rubber  tube  and 
pinch-cock,  be  partially  filled,  by  suction,  with  a  solution  of 
copper  sulphate,  and  then  immersed  in  a  solution  of  potassium 
ferrocyanide,  a  thin  film  of  copper  ferrocyanide  will  be  formed 
at  the  junction  of  the  two  solutions.  When  the  film  has  once 
been  formed,  further  precipitation  of  copper  ferrocyanide  will 
cease,  the  solutions  on  either  side  of  the  film  remaining  clear. 
Traube  showed  that  this  membrane  is  semi-permeable.  He  also 
showed  that  a  number  of  other  gelatinous  precipitates  possess 
the  property  of  semi-permeability.  A  membrane  formed  in  the 
above  manner  is  easily  ruptured  and  is  wholly  inadequate  for 
quantitative  or  even  qualitative  experiments.  Pfeffer  f  devised 
a  method  for  strengthening  the  membrane.  By  depositing  the 
precipitate  in  the  walls  of  a  porous  clay  cup,  the  area  of  unsup- 
ported membrane  is  greatly  diminished  and  its  resisting  power 
correspondingly  increased.  Pfeffer  directs  that  the  cup  to  be 
used  for  this  purpose  must  be  throughly  washed,  and  its  walls 
allowed  to  become  completely  permeated  with  water.  The  cup 
is  then  filled  to  the  top  with  a  solution  of  copper  sulphate,  con- 
taining 2.5  grams  per  liter,  and  allowed  to  stand  for  several  hours 
in  a  solution  of  potassium  ferrocyanide,  containing  2.1  grams 
per  liter.  The  two  solutions  diffuse  through  the  walls  of  the  cup 
and  on  meeting,  deposit  a  thin  membrane  of  copper  ferrocyanide. 
When  precipitation  is  complete,  the  cup  is  thoroughly  washed 
and  soaked  in  water.  The  cup  is  then  filled  to  the  top  with  a 
solution  of  cane  sugar,  and  a  rubber  stopper,  fitted  with  a  long 
glass  tube  of  narrow  bore,  is  inserted,  care  being  taken  to  exclude 
air-bubbles.  The  stopper  is  then  made  fast  with  a  suitable 

*  Archiv.  fur  Anat.  und  Physiol.,  p.  87  (1867). 
f  Osmotische  Untersuchungen,  Leipzig,  1877. 


DILUTE  SOLUTIONS  AND  OSMOTIC  PRESSURE         189 


cement,  and  the  cup  completely  immersed  in  a  beaker  of  water. 
The  completed  apparatus  is  shown  in  Fig.  58.  If  the  formation 
of  the  membrane  has  been  successful,  the  level  of  the  liquid  in  the 
vertical  glass  tube  will  slowly  rise  and  will  eventually  attain  a 
height  of  several  meters.  If  the  mem- 
brane is  sufficiently  strong  and  no  leaks 
develop,  the  passage  of  water  through  the 
membrane  will  continue  until  the  hydro- 
static pressure  of  the  column  of  liquid  in 
the  tube  is  great  enough  to  overcome  the 
tendency  of  the  water  to  force  its  way 
into  the  sugar  solution.  As  a  general 
rule,  the  membrane  becomes  ruptured  be- 
fore equilibrium  is  attained. 

Measurement  of  Osmotic  Pressure. 
The  first  direct  measurements  of  osmotic 
pressure  were  made  by  Pfeffer.  His  ex- 
periments deserve  brief  consideration, 
since  the  results  obtained  furnish  the 
basis  of  the  modern  theory  of  solution. 
The  cell  used  was  similar  to  that  described 
above,  but  instead  of  employing  a  ver- 
tical glass  tube  as  a  manometer,  the  cup 
was  connected,  as  shown  in  Fig.  59,  with 
a  closed  mercury  manometer.  The  sub- 
stitution of  the  closed  for  the  open  man- 
ometer is  necessitated  by  the  fact,  that 
with  an  open  manometer  so  much  water  entered  the  cell  that 
the  concentration  of  the  solution  became  appreciably  diminished, 
and  the  pressure  actually  measured  corresponded  to  a  solu- 
tion of  smaller  concentration  than  that  introduced  into  the  cell. 
With  the  closed  manometer,  when  a  trace  of  water  has  entered 
the  cell,  sufficient  pressure  is  developed  to  prevent  the  further  en- 
trance of  more  water.  Pfeffer  calculated  that  with  a  cell,  the 
capacity  of  which  was  16  cc.,  the  volume  of  water  entering  before 
equilibrium  was  attained,  did  not  exceed  0.14  cc.  In  his  experi- 
ments, Pfeffer  determined  the  density  of  the  cell  contents  before  and 


Fig.  58. 


190 


THEORETICAL  CHEMISTRY 


after  measurement  of  the  osmotic  pressure,  and  corrected  for  any 
change  in  concentration.  With  this  apparatus  he  made  numer- 
ous measurements  of  th«  osmotic 
pressures  of  different  solutions,  the 
entire  apparatus  being  immersed  in 
a  constant-temperature  bath.  With 
solutions  of  cane  sugar  he  obtained 
the  results  given  in  the  accompany- 
ing table,  where  C  denotes  the  per- 
centage concentration  of  the  solution, 
and  P  the  corresponding  osmotic 
pressure,  expressed  in  centimeters  of 
mercury.  The  temperature  varied 
from  13.5°  C.  to  14°.7  C. 


C 

P 

P/C 

1 

53.5 

53.5 

2 

101.6 

50.8 

4 

208.2 

52.0 

6 

307.5 

51.2 

It  is  evident  from  these  results, 
that  the  osmotic  pressure  is  propor- 
tional to  the  concentration  of  the 
solution,  since  P/C  is  approximately 
constant.  The  deviations  from  con- 
stancy in  the  ratio  of  pressure  to 
concentration  may  be  ascribed  to 
experimental  errors,  since  the  dif- 
ficulties involved  in  these  measure- 
ments are  very  great.  Pfeffer  also 
Fig.  59.  studied  the  influence  of  temperature 

on  osmotic   pressure,  and   showed 

that  as  the  temperature  is  raised  the  pressure  increases.  The 
following  table  gives  his  results  for  a  1  per  cent  solution  of  cane 
sugar. 


DILUTE  SOLUTIONS  AND   OSMOTIC  PRESSURE 


191 


Temperature. 

Osmotic  Pressure. 

6°.  8 

cm. 

50.5 

13°.  2 

52.1 

14°.  2 

53.1 

22°.  0 

54.8 

36°.  0 

56.7 

Osmotic  Pressure  and  the  Nature  of  the  Membrane.  Pfeffer 
also  studied  the  effect  of  the  nature  of  the  membrane  on  osmotic 
pressure.  In  addition  to  copper  ferrocyanide,  he  used  membranes 
of  calcium  phosphate  and  Prussian  blue.  His  results  seemed  to 
indicate  that  the  magnitude  of  the  osmotic  pressure  developed, 
was  dependent  upon  the  nature  of  the  membrane  used. 

The  variations  observed  have  since  been  shown  to  have  been 
due  to  leakage  of  the  calcium  phosphate  and  Prussian  blue  mem- 
branes, the  copper  ferrocyanide  membrane  being  the  only  one 
which  was  capable  of  withstanding  the  pressure.  Ostwald  *  has 
devised  an  ingenious  theoretical  demonstration  of  the  fact  that 
osmotic  pressure  must  be  independent  of  the  nature  of  the  mem- 
brane employed  in  measuring  it.  Let  A  and  B,  in  Fig.  60,  repre- 

A  B 


Fig.  60. 

sent  two  different  semi-permeable  membranes  placed  in  a  glass 
tube  of  wide  bore.  Let  us  imagine  the  space  between  the  two 
membranes  to  be  filled  with  a  solution,  and  the  tube  immersed 
in  a  vessel  of  water.  If  the  osmotic  pressures  developed  at  A 
and  B  are  pi  and  p2  respectively,  and  p2  is  less  than  pi,  then 
water  will  pass  through  A  until  the  pressure  pi  is  reached.  Since 
the  pressure  at  B  only  reaches  the  value  p2,  however,  the  pres- 
sure pi  can  never  be  attained,  and  a  steady  stream  of  water  from 
A  to  B,  under  the  pressure  p\  —  pz,  will  result.  This,  however, 
would  be  a  perpetual  motion,  and  since  this  is  impossible,  the 
osmotic  pressures  at  the  two  membranes  must  be  the  same. 
*  Lehrb.  d.  allg.  Chem.,  I.,  p.  662.. 


192 


THEORETICAL  CHEMISTRY 


Theoretical  Value  of  Osmotic  Pressure.  The  physico-chem- 
ical significance  of  Pfeffer's  results  was  first  perceived  by  van't 
Hoff.*  In  a  remarkably  brilliant  paper,  he  pointed  out  the 
existence  of  a  striking  parallelism  between  the  properties  of  gases 
and  the  properties  of  dissolved  substances. 

We  have  already  called  attention  to  the  analogy  between  osmotic 
pressure  and  gas  pressure:  we  now  proceed  to  trace  the  connec- 
tion between  osmotic  pressure,  volume  and  temperature,  as  first 
pointed  out  by  van't  Hoff.  Pfeffer's  experiments  showed  that  at 
constant  temperature,  the  ratio,  P/C,  is  constant  for  any  one 
solute.  Since  the  concentration  varies  inversely  as  the  volume 
in  which  a  definite  amount  of  solute  is  dissolved,  we  obtain,  by 
substituting  l/V  for  C,  the  equation,  PV  =  constant,  which  is 
plainly  the  analogue  of  the  familiar  equation  of  Boyle  for  gases. 
An  examination  of  Pfeffer's  data  for  osmotic  pressures  at  differ- 
ent temperatures,  convinced  van't  Hoff  that  the  law  of  Gay- 
Lussac  is  also  applicable  to  solutions. 

In  the  following  table,  the  osmotic  pressures  in  atmospheres  for 
a  1  per  cent  solution  of  cane  sugar  at  different  temperatures  are 
recorded,  together  with  the  pressures  calculated  on  the  assump- 
tion that  the  osmotic  pressure  is  directly  proportional  to  the 
absolute  temperature. 


Temperature. 

P  (obs.). 

P  (calc.). 

6°.  8 

0.664 

0.665 

13°.  7 

0.691 

0.681 

15°.  5 

0.684 

0.686 

22°.  0 

0.721 

0.701 

32°.  0 

0.716 

0.725 

36°.  0 

0.746 

0.735 

Since  the  laws  of  Boyle  and  Gay-Lussac  are  both  applicable, 
we  may  write  an  equation  for  dilute  solutions  corresponding  to 
that  already  derived  for  gases,  or 

PV  =  R'T, 

in  which  P  is  the  osmotic  pressure  of  a  solution  containing  a  defi- 
*  Zeit.  phys.  Chem.,  i,  481  (1887). 


DILUTE  SOLUTIONS  AND   OSMOTIC  PRESSURE         193 

nite  weight  of  solute  in  the  volume,  V,  of  solution,  T  being  the 
absolute  temperature  of  the  solution  and  Rr  a  constant  corre- 
sponding to  the  molecular  gas  constant. 

The  molecular  gas  constant  R  has  already  been  evaluated  and 
has  been  found  to  be  equal  to  0.0821  liter-atmosphere. 

Making  use  of  Pfeffer's  data,  van't  Hoff  calculated  the  value 
of  R'  in  the  above  equation,  in  the  following  manner :  the  osmotic 
pressure  of  a  1  per  cent  solution  of  cane  sugar  at  0°  C.  is  0.649 
atmosphere,  and  since  the  concentration  of  the  solution  is  1  per 
cent,  the  volume  of  solution  containing  1  mol  of  sugar,  will  be 
34,200  cc.  or  34.2  liters.  Substituting  these  values  in  the  equa- 
tion, we  have 

D,      PV      0.649  X  34.2 

R'  =  -™-  = 273 =  0-0813  liter-atmos., 

a  value  which  is  nearly  the  same  as  that  of  the  molecular  gas 
constant,  R.  The  equality  of  R  and  R'  leads  to  a  conclusion  of 
the  greatest  importance,  as  was  pointed  out  by  van't  Hoff,  viz., 
"  the  osmotic  pressure  exerted  by  any  substance  in  solution  is  the  same 
as  it  would  exert  if  present  as  a  gas  in  the  same  volume  as  that  occupied 
by  the  solution,  provided  that  the  solution  is  so  dilute  that  the  volume 
occupied  by  the  solute  is  negligible  in  comparison  with  that  occupied 
by  the  solvent. "  It  should  be  remembered  that  we  are  not  justified 
in  concluding  from  this  proposition  of  van't  Hoff,  that  osmotic 
pressure  and  gaseous  pressure  have  a  common  origin.  While 
the  origin  of  osmotic  pressure  may  be  kinetic,  it  is  also  conceivable 
that  it  may  result  from  the  mutual  attraction  of  solvent  and 
solute,  or  that  it  may  bear  some  relation  to  the  surface  ten- 
sion of  the  solution.  Up  to  the  present  time  no  wholly  satis- 
factory explanation  of  the  cause  of  osmotic  pressure  has  been 
advanced. 

Just  as  1  mol  of  gas  at  0°  C.  and  760  mm.  pressure  occupies  a 
volume  of  22.4  liters,  so  when  1  mol  of  a  substance  is  dissolved 
and  the  solution  diluted  to  22.4  liters  at  0°  C.,  it  will  exert  an 
osmotic  pressure  of  1  atmosphere.  In  other  words,  molar  weights, 
or  quantities  proportional  to  molar  weights,  of  different  substances, 
when  dissolved  in  equal  volumes  of  the  same  solvent  exert  the  same 


194  THEORETICAL  CHEMISTRY 

osmotic  pressure.  If  we  deal  with  n  mols  of  solute  instead  of  1  mol, 
the  general  equation  becomes 

PV  =  nRT. 

But  n  =  g/M,  where  g  is  the  number  of  grams  of  solute  per  liter, 
and  M  is  its  molecular  weight.  Substituting  in  the  preceding 
equation,  we  have 

PV  =  g/M-RT, 
or, 

M-gET 
"PV 

Since  P,  V,  g,  R  and  T  are  all  known,  M  can  be  calculated.  The 
direct  measurement  of  the  osmotic  pressure  of  a  solution  does  not 
afford  a  practical  method  for  the  determination  of  the  molecular 
weight  of  dissolved  substances,  because  of  the  experimental 
difficulties  involved  and  the  time  required  for  the  establishment 
of  equilibrium.  There  are  other  and  simpler  methods  for  deter- 
mining molecular  weights  in  solution,  based  upon  certain  proper- 
ties of  solutions  which  are  proportional  to  their  respective  osmotic 
pressures. 

Recent  Work  on  the  Direct  Measurement  of  Osmotic  Pressure. 
It  is  only  within  the  past  decade  that  the  investigations  of  Pfeffer 
have  been  confirmed  and  extended  by  elaborate  and  systematic 
experiments  on  the  direct  measurement  of  osmotic  pressure. 
Morse  and  his  co-workers,*  while  employing  a  method  essentially 
the  same  as  that  of  Pfeffer,  have,  as  the  result  of  much  patient 
labor,  brought  the  apparatus  to  such  a  high  state  of  perfection, 
that  the  experimental  errors  are  now  estimated  to  affect  only 
the  second  place  of  decimals  in  the  numerical  data  expressing 
osmotic  pressures  in  atmospheres.  The  most  important  of  the 
improvements  introduced  by  Morse  are  the  following: —  (1)  the 
improvement  of  the  quality  of  the  membrane;  (2)  the  improve- 
ment of  the  connection  between  the  cell  and  the  manometer, 
and  (3)  the  improvement  of  the  means  of  accurately  measuring 

*  Am.  Chem.  Jour.,  26,  80  (1901);  34,  1  (1905);  36,  1,  39  (1906);  37,  324, 
425,  558  (1907);  38,  175  (1907);  39,  667  (1908);  40,  1,  194  (1908);  41,  1,  257 
(1909). 


DILUTE  SOLUTIONS  AND   OSMOTIC   PRESSURE 


195 


the  pressure.  The  membrane  of  copper 
ferrocyanide  is  deposited  electrolyti- 
cally.  After  thorough  washing  and 
soaking  in  water,  the  porous  cup,  made 
from  specially  prepared  clay,  is  filled 
with  a  solution  of  potassium  ferrocy- 
anide and  immersed  in  a  solution  of  cop- 
per sulphate.  An  electric  current  is 
then  passed  from  a  copper  electrode 
in  the  solution  of  copper  sulphate,  to 
a  platinum  electrode  immersed  in  the 
solution  of  potassium  ferrocyanide. 
This  drives  the  copper  and  ferrocy- 
anide ions  toward  each  other,  and  the 
membrane  of  copper  ferrocyanide  is 
thus  formed  in  the  walls  of  the  cup. 
The  passage  of  the  current  is  continued 
until  the  electrical  resistance  reaches  a 
value  of  about  100,000  ohms.  The 
cell  is  then  rinsed,  and  soaked  in  water 
for  several  hours,  and  then  the  electro- 
lytic treatment  is  repeated  until  the 
electrical  resistance  attains  a  maximum 
value.  A  solution  of  cane  sugar  is  now 
introduced  into  the  cell,  which  is  con- 
nected with  the  manometer  and  im- 
mersed in  water.  When  the  pressure 
has  attained  its  maximum  value,  the 
apparatus  is  dismantled  and  the  cell, 
after  thorough  washing  and  soaking  in 
water,  is  again  subjected  to  the  electro- 
lytic process  of  membrane  forming.  In 
this  way  the  weak  places  in  the  mem- 
brane which  may  have  yielded  to  the 
high  pressure,  can  be  repaired,  and  by  Fig.  61. 

continued  repetition  of  this  treatment 
the  membrane  can  ultimately  be  brought  to  its  maximum  power 


196 


THEORETICAL  CHEMISTRY 


of  resistance.  A  sketch  of  the  Morse  apparatus  is  shown  in  Fig. 
61.  A  description  of  the  details  of  this  apparatus  lies  beyond  the 
scope  of  this  book.  The  results  of  the  work  of  Morse  and  his 
students  are  of  the  highest  importance.  The  osmotic  pressures 
of  solutions  of  cane  sugar  and  dextrose  have  been  shown  to  be 
proportional  to  the  respective  concentrations,  provided  the  con- 
centration is  referred  to  unit  volume  of  solvent  instead  of  unit 
volume  of  solution.  Thus  in  their  experiments,  the  solutions  were 
made  up  containing  from  0.1  to  1.0  mol  of  solute  in  1000  grams  of 
water.  Morse  calls  such  solutions  weight-normal  solutions  in  con- 
trast to  volume-normal  solutions,  in  which  1  mol  or  a  fraction  of 
a  mol  of  solute  is  dissolved  in  water  and  the  solution  diluted  to  1 
liter.  The  following  data  taken  from  the  work  of  Morse,  shows 
that  when  concentration  is  expressed  on  the  weight-normal  basis, 
there  is  direct  proportionality  between  osmotic  pressure  and  con- 
centration. The  figures  refer  to  solutions  of  dextrose  at  10°  C. 


Molar  Concentration. 

Osmotic  Pressure. 

Per  1000  gm. 
Water. 

Per  Liter  of 
Solution. 

In  Atmos. 

Relative  to  First 
as  Unity. 

0.1 

0.099 

2.39 

1.00 

0.2 

0.196 

4.76 

1.99 

0.5 

0.474 

11.91 

4.98 

1.0 

0.901 

23.80 

9.96 

Morse  and  his  co-workers  also  conclude  from  their  experiments 
at  temperatures  ranging  from  0°  C.  to  25°  C.,  that  the  temper- 
ature coefficients  of  osmotic  pressure  and  gas  pressure  are  prac- 
tically identical.  In  other  words,  their  results  confirm  the 
conclusions  of  van't  Hoff,  that  the  law  of  Gay-Lussac  is  applicable 
to  solutions.  The  results  of  the  experiments  of  Morse  are  of 
special  interest  in  connection  with  the  proposition  of  van't  Hoff, 
that  the  osmotic  pressure  of  a  dilute  solution  is  the  same  as  that 
which  the  solute  would  exert  if  it  were  gasified  at  the  same  temper- 
ature and  occupied  the  same  volume  as  the  solution.  The  data 
in  the  following  table  is  taken  from  the  work  of  Morse  on  solutions 


DILUTE  SOLUTIONS  AND   OSMOTIC  PRESSURE 


197 


of  cane  sugar  at  15°  C.  In  addition  to  the  observed  osmotic 
pressures,  the  table  contains  the  corresponding  gas  pressures, 
calculated  (1)  on  the  assumption  that  the  solute  when  gasified 
occupies  the  same  volume  as  the  solution  (proposition  of  Van't 
Hoff),  and  (2)  on  the  assumption  that  it  occupies  the  same  volume 
as  the  solvent  alone. 


Molar  Concentration. 

Osmotic  Pressure  in  Atmos. 

Per  1000  gm.  of 
Water. 

Per  Liter  of 
Solution. 

Obs. 

Calc.  (a). 

Calc.  (b). 

0.1 

0.098 

2.48 

2.30 

2.35 

0.2 

0.192 

4.91 

4.51 

4.70 

0.4 

0.369 

9.78 

8.67 

9.40 

0.6 

0.533 

14.86 

12.51 

14.08 

0.8 

0.684 

20.07 

16.07 

18.79 

1.0 

0.825 

25.40 

19.38 

23.49 

The  calculated  pressures,  recorded  in  the  last  column,  are  in 
much  closer  agreement  with  the  observed  osmotic  pressures,  than 
are  the  calculated  pressures,  recorded  in  the  fourth  column  of  the 
table.  The  proposition  of  van't  Hoff  should  then  be  modified 
to  read  as  follows :  —  A  dissolved  substance  in  dilute  solution  exerts 
an  osmotic  pressure  equal  to  that  which  it  would  exert  if  it  were  gas- 
ified at  the  same  temperature,  and  the  volume  of  the  gas  were  reduced 
to  that  of  the  solvent  in  the  pure  state.  The  investigations  of  Morse 
and  his  co-workers  may  be  summarized  thus: —  (1)  the  law  of 
Boyle  is  applicable  to  dilute  solutions,  provided  the  concentration 
is  referred  to  1000  grams  of  solvent  and  not  to  1  liter  of  solution; 
(2)  the  law  of  Gay-Lussac  is  also  applicable  to  dilute  solutions, 
that  is,  the  temperature  coefficients  of  osmotic  pressure  and  gas 
pressure  are  equal,  and  (3)  the  small  departures  from  the  theo- 
retical values  of  the  osmotic  pressures  may  be  traced  to  hydration 
of  the  solute. 

Direct  measurements  of  the  osmotic  pressure  of  concentrated 
solutions  of  cane  sugar,  dextrose  and  mannite  have  been  made  by 
the  Earl  of  Berkeley  and  E.  G.  J.  Hartley.*  The  method  em- 

*  Proc.  Roy.  Soc.,  73,  436  (1904);  Trans.  Roy.  Soc.  A.,  206,  481  (1906). 


198  THEORETICAL  CHEMISTRY 

ployed  by  these  investigators  is  slightly  different  from  that  of 
Pfeffer  or  Morse;  the  tendency  of  water  to  pass  through  the 
semi-permeable  membrane  is  offset  by  the  application  of  a  counter 
pressure  to  the  solution.  A  membrane  of  copper  ferrocyanide 
is  deposited  electrolytically  very  near  the  outer  surface  of  a  tube 
of  porous  porcelain.  This  tube  is  placed  co-axially  within  a  large 
cylindrical  vessel  of  gun  metal,  an  absolutely  tight  joint  between 
the  two  being  secured  by  an  ingenious  system  of  dermatine  rings 
and  clamps.  The  open  ends  of  the  porcelain  tube  are  closed  by 
rubber  stoppers  fitted  with  capillary  tubes  bent  at  right  angles, 
one  of  the  latter  being  provided  with  a  glass  stop-cock.  When  a 
determination  of  osmotic  pressure  is  to  be  made,  the  apparatus  is 
placed  in  a  horizontal  position  and  water  is  introduced  into  the 
porcelain  tube,  completely  filling  it  and  the  connecting  capillary 
tubes  up  to  a  certain  level.  The  gun  metal  vessel  is  then  filled 
with  the  solution,  and  connected  with  an  auxiliary  apparatus  by 
means  of  which  a  gradually  increasing  hydrostatic  pressure  can 
be  applied.  If  no  pressure  is  applied  to  the  solution,  water  will 
pass  through  the  semi-permeable  membrane  into  the  solution,  and 
the  level  of  the  water  in  the  capillary  tubes  will  fall.  In  carrying 
out  a  measurement,  therefore,  as  soon  as  the  solution  is  introduced 
into  the  gun  metal  vessel,  hydrostatic  pressure  is  applied,  the  mag- 
nitude of  the  pressure  being  so  adjusted  as  to  counterbalance  the 
osmotic  pressure  of  the  solution.  The  level  of  the  water  in  the 
capillary  tubes  serves  to  indicate  the  relative  magnitudes  of 
the  osmotic  and  hydrostatic  pressures.  When  the  level  of  the 
water  in  the  capillary  tubes  remains  constant,  the  two  pressures 
are  in  equilibrium.  The  following  are  the  values  of  the  equilibrium 
pressures  of  solutions  of  cane  sugar,  dextrose  and  mannite  at  0°  C. 
It  must  be  remembered  that  when  the  two  pressures  are  in  equi- 
librium, there  is  always  a  pressure  of  one  atmosphere  on  the  solvent. 
As  will  be  seen  the  pressures  developed  in  the  more  concen- 
trated solutions  are  enormous  and  it  is  a  surprising  fact,  that  even 
in  cases  where  the  highest  pressures  were  measured,  hardly  a 
trace  of  sugar  was  found  in  the  pure  solvent,  the  membrane 
retaining  its  property  of  semi-permeability  throughout  the  entire 
range  of  pressures.  The  figures  in  the  third  column  are  calculated 


DILUTE  SOLUTIONS  AND  OSMOTIC  PRESSURE 


199 


CANE  SUGAR. 


Osmotic  Pressure  in  Atmospheres. 

Cono.  gm.  per 
Liter. 

Obs. 

Calc. 

180.1 

13.95 

13.95 

300.2 

26.77 

28.74 

420.3 

43.97 

32.55 

540.4 

67.51 

41.85 

660.5 

100.78 

51.16 

750.6 

133.74 

58.14 

DEXTROSE. 


Osmotic  Pressure  in  Atmospheres. 

Cone.  gm.  per 
Liter. 

Obs. 

Calc. 

99.8 

199.5 
319.2 
448.6 
548.6 

13.21 

29.17 
53.19 

87.87 
121  .  18 

13.21 
26.41 
42.25 
59.28 
72.61 

MANNITE. 


Osmotic  Pressure  in  Atmospheres. 

Cone.  gm.  per 
Liter. 

Obs. 

Calc. 

100 
110 
125 

13.1 
14.6 
16.7 

13.1 
14.4 
16.4 

on  the  assumption  that  there  is  direct  proportionality  between 
osmotic  pressure  and  concentration.  It  is  apparent  that  in  every 
case  the  observed  osmotic  pressure  is  greater  than  the  calculated. 
Even  when  the  concentrations  are  expressed  on  the  weight-normal 


200 


THEORETICAL  CHEMISTRY 


basis,  as  recommended  by  Morse,  the  osmotic  pressure  increases 
more  rapidly  than  the  concentration. 

This  is  well  shown  in  the  accompanying  diagram,  Fig.  62,  due 
to  the  Earl  of  Berkeley.  In  this  diagram,  the  osmotic  pressures 
of  solutions  of  cane  sugar  are  plotted  against  concentrations,  curve  A 
representing  the  actually  observed  osmotic  pressures;  curve  C 


120 


0  100  200  300  400  500  600  700 

Grams  Cane  Sugar  per  liter  of  Solution 

Fig.  62. 

being  traced  on  the  assumption  that  osmotic  pressure  may  be 
calculated  from  the  equation,  PV  =  RT,  where  V  denotes  the 
volume  of  solvent  containing  1  mol  of  cane  sugar;  and  curve  B, 
a  straight  line,  being  drawn  on  the  assumption  that  osmotic 
pressure  may  be  calculated  from  the  equation,  PV  =  RT,  where 
V  represents  the  volume  of  solution  containing  1  mol. 

While  the  theoretical  and  observed  values  of  the  osmotic  pres- 


DILUTE  SOLUTIONS  AND  OSMOTIC   PRESSURE          201 

sure  are  approximately  equal  in  the  more  dilute  solutions,  it  is 
obvious  that  the  observed  values  of  the  osmotic  pressure  of  the 
concentrated  solutions  are  always  greater  than  the  calculated 
values,  even  when  the  calculation  is  made  on  the  assumption  that 
V  in  the  equation,  PV  =  RT,  is  the  volume  of  the  solvent.  The 
abnormally  high  osmotic  pressures  observed  by  the  Earl  of 
Berkeley  have  been  discussed  by  Callendar  *  who  suggests  hydra- 
tion  of  the  solute  as  a  probable  cause. 

He  shows,  that  if  5  molecules  of  water  are  assumed  to  be  asso- 
ciated with  each  molecule  of  cane  sugar  in  the  most  concentrated 
solutions  studied  by  the  Earl  of  Berkeley,  the  discrepancy  between 
the  observed  and  calculated  values  of  the  osmotic  pressure  dis- 
appears. 

Comparison  of  Osmotic  Pressures.  Although  the  difficulties 
involved  in  the  direct  determination  of  osmotic  pressure  are 
many,  these  can  be  avoided  by  the  employment  of  one  of  several 
indirect  methods  which  have  been  devised  for  the  comparison  of 
osmotic  pressures.  All  of  these  methods  depend  upon  the  exchange 
of  water  which  occurs  when  two  solutions  are  separated  by  a 
semi-permeable  membrane.  The  movement  of  the  water  will 
always  be  in  such  a  direction  as  to  tend  to  equalize  the  osmotic 
pressures  on  opposite  sides  of  the  membrane,  or,  in  other  words, 
the  transfer  of  water  will  take  place  from  the  solution  with  the 
lesser  osmotic  pressure  to  the  solution  with  the  greater  osmotic 
pressure. 

The  Plasmolytic  Method.  In  this  method,  solutions  of  various 
substances  are  prepared,  the  concentration  of  each  being  such 
that  its  osmotic  pressure  is  the  same  as  that  of  a  particular  plant 
cell.  Obviously  the  osmotic  pressures  of  all  of  these  solutions 
must  be  equal :  such  solutions  are  said  to  be  isotonic  or  isosmotic. 
The  plasmolytic  method  for  the  comparison  of  osmotic  pressures 
was  developed  by  the  Dutch  botanist,  De  Vries.f  This  method 
depends  upon  the  shrinking  or  swelling  of  the  protoplasmic  sac 
of  plant  cells  when  they  are  immersed  in  a  solution  whose  osmotic 

*  Proc.  Roy.  Soc.  A.,  80,  466  (1908). 

f  Jahrb.  wiss.  Botanik.,  14,  427  (1884);  Zeit.  phys.  Chem.,  2,  415  (1888);  3, 
103  (1889). 


202  THEORETICAL  CHEMISTRY 

pressure  differs  from  that  of  their  own  sap.  De  Vries  found  that 
the  cells  of  Tradescantia  discolor,  Curcuma  rubricaulis,  and  Begonia 
manicata  fulfil  the  necessary  conditions,  viz.;  the  cell  walls  are 
strong  and  resist  alteration  when  immersed  in  solutions,  the  cells 
are  readily  permeable  to  water,  and  the  cell  contents  are  colored, 
thus  enabling  the  slightest  contraction  or  expansion  to  be  de- 
tected. The  "cell  walls  are  lined  on  the  inside  with  a  thin,  elastic, 
semi-permeable  membrane  which  encloses  the  colored  contents 
of  the  cell.  The  content  of  the  cell  consists  of  an  aqueous  solu- 
tion of  several  substances,  among  which  may  be  mentioned 
glucose,  potassium  and  calcium  malate,  together  with  coloring 
matter.  The  osmotic  pressure  of  the  cell  contents  ranges  from 
four  to  six  atmospheres.  The  semi-permeable  membrane  expands 
when  the  contents  of  the  cell  increases  and  contracts  when  the 
contents  diminishes.  In  making  a  comparison  of  osmotic  pres- 
sures by  this  method,  tangential  sections  are  cut  from  the  under 
side  of  the  mid-rib  of  the  leaf  of  one  of  the  above  plants,  e.g., 
Tradescantia  discolor,  and  are  placed  in  the  solution  whose  osmotic 
pressure  it  is  desired  to  compare  with  that  of  the  cell  contents. 
The  cells  are  then  observed  under  the  microscope,  any  decrease 
in  pressure  below  the  normal  resulting  in  a  detachment  of  the 
semi-permeable  membrane  from  one  or  more  points  of  the  cell 
wall.  This  contraction  always  occurs  when  the  cells  are  im- 
mersed in  a  concentrated  solution,  the  phenomenon  being  termed 
plasmolysis.  When  the  solution  in  which  the  cells  are  placed 
has  a  lower  osmotic  pressure  than  the  cell  contents,  no  visible 
effect  is  produced,  the  increased  pressure  within  the  cell  simply 
forcing  the  membrane  closer  to  the  rigid  cell  walls.  By  starting 
with  a  concentrated  solution,  the  osmotic  pressure  of  which  is 
greater  than  that  of  the  cell,  and  gradually  diluting  it,  a  concen- 
tration will  ultimately  be  reached  at  which  the  elastic  membrane 
will  just  completely  fill  the  cell.  This  solution  is  isotonic  with 
the  cell  contents.  In  this  method  the  very  reasonable  assump- 
tion is  made  that  all  of  the  cells  have  the  same  osmotic  pressure, 
any  differences  which  might  have  existed  having  equalized  them- 
selves in  the  living  plant.  The  microscopic  appearance  of  cells 
of  Tradescantia  discolor  when  immersed  in  different  solutions  is 


DILUTE  SOLUTIONS  AND  OSMOTIC  PRESSURE 


203 


shown  in  Fig.  63.  The  appearance  of  the  normal  cell  when  im- 
mersed in  water  or  in  a  solution  whose  osmotic  pressure  is  less 
than  that  of  the  cell  contents,  is  shown  in  A .  When  the  cell  is 
immersed  in  a  0.22  molar  solution  of  cane  sugar  it  appears  as  in 
B,  this  solution  having  a  greater  osmotic  pressure  than  the  cell 
contents.  When  the  cell  is  immersed  in  a  molar  solution  of 


Fig.  63. 

potassium  nitrate,  there  is  marked  plasmolysis,  as  shown  in  C. 
De  Vries  determined  the  concentrations  of  a  large  number  of 
solutions  which  were  isotonic  with  the  cell  contents.  He  ex- 
pressed his  results  in  terms  of  the  isotonic  coefficient,  which  he 
defined  as  the  reciprocal  of  the  molar  concentration.  The  iso- 
tonic coefficient  of  potassium  nitrate  was  taken  equal  to  3.  A 
few  of  De  Vries'  results  are  given  in  the  following  table. 


Substance. 

Formula. 

Isotonic 
Coefficient. 

Glycerol  . 

C3H8O8 

1.78 

Glucose  

C6H12O6 

1.81 

Cane  sugar  

1.88 

Malic  acid  

C4H6O5 

1.98 

Tartaric  acid  

C4H6O6 

2.02 

Citric  acid  

C6H8O7 

2.02 

Potassium  nitrate 

KNO3 

3.00 

Magnesium  chloride  

MgCl2 

4.33 

204  THEORETICAL  CHEMISTRY 

De  Vries  applied  the  plasmolytic  method  to  the  determination 
of  the  molecular  weight  of  raffinose.  At  that  time  there  was 
considerable  uncertainty  as  to  the  correct  formula  of  crystallized 
ramnose,  three  different  formulas,  all  consistent  with  the  results 
of  analysis,  having  been  proposed  as  follows:  —  CisH^Oie- 
5  H2O,  Ci2H22Oii.3  H2O,  and  C36H64O32.  De  Vries  found  that  a 
3.42  per  cent  solution  of  cane  sugar  was  isotonic  with  a  5.96  per 
cent  solution  of  ramnose.  Letting  the  unknown  molecular 
weight  of  ramnose  be  represented  by  M,  then 

3.42  :5.96  ::342:M. 

Solving  the  proportion  we  have,  M  =  596.  This  result  has  since 
been  confirmed  by  purely  chemical  methods,  and  the  formula, 
Ci8H32Oi6.5  H20,  the  molecular  weight,  of  which  is  594,  is  thus 
established. 

The  Blood  Corpuscle  Method.  The  red  blood  corpuscle  is  a 
cell,  the  contents  of  which  is  enclosed  by  a  thin  elastic  semi- 
permeable  membrane.  Unlike  the  plant  cells,  there  is  no  resistant 
cell  wall  to  give  support  to  the  membrane,  so  that  when  red  blood 
corpuscles  are  immersed  in  water  they  at  first  swell,  owing  to  the 
osmotic  pressure  developed,  and  finally  burst.  When  the  mem- 
brane is  ruptured,  the  coloring  matter  of  the  cell,  the  haemoglobin, 
escapes  and  the  water  acquires  a  deep  red  color. 

Advantage  of  this  behavior  of  red  blood  corpuscles  was  taken 
by  Hamburger  *  for  the  comparison  of  osmotic  pressures.  He 
found  that  when  a  1.04  per  cent  solution  of  potassium  nitrate 
is  added  to  the  defibrinated  blood  of  a  bullock,  the  corpuscles 
will  settle  completely  to  the  bottom,  while  the  supernatant  liquid 
will  remain  clear.  On  the  other  hand,  if  a  0.96  per  cent  solution 
of  potassium  nitrate  is  used,  the  corpuscles  will  not  settle  and  the 
supernatant  liquid  becomes  colored.  If  more  dilute  solutions  of 
potassium  nitrate  are  used,  the  solution  acquires  a  still  deeper 
color.  By  careful  adjustment,  a  concentration  of  potassium 
nitrate  can  be  found  in  which  the  red  blood  corpuscles  will  just 
settle.  In  like  manner,  the  concentration  of  solutions  of  other 
substances  can  be  so  adjusted  as  to  cause  the  precipitation  of  the 
*  Zeit.  phys.  Chem.,  6,  319  (1890). 


DILUTE   SOLUTIONS  AND   OSMOTIC  PRESSURE          205 

corpuscles.  These  solutions  are  isotonic.  Without  going  into 
details,  it  may  be  said  that  the  isotonic  coefficients  obtained  by 
Hamburger,  agree  well  with  those  obtained  by  the  plasmolytic 
method. 

The  Haematocrit  Method.  In  this  method  developed  by 
Hedin,*  advantage  is  again  taken  of  the  properties  of  red  blood 
corpuscles.  As  has  already  been  stated,  when  red  blood  cor- 
puscles are  immersed  in  solutions  of  gradually  diminishing 
concentration  of  the  same  solute,  they  continue  to  swell  and  ulti- 
mately the  semi-permeable  envelope  bursts.  On  the  other  hand, 
when  the  corpuscles  are  immersed  in  solutions  of  gradually  increas- 
ing concentration,  they  shrink,  owing  to  the  transfer  of  water 
from  the  corpuscles.  It  is  apparent  that  there  must  be  a  certain 
concentration  for  each  solute  which  will  cause  no  change  in  the 
volume  of  the  corpuscles.  To  determine  this  concentration,  use 
is  made  of  an  instrument  known  as  an  hcematocrit.  This  is 
simply  a  graduated  thermometer-tube  which  may  be  attached  to 
the  spindle  of  a  centrifugal  machine.  When  the  spindle  is  re- 
volved at  high  speed,  the  corpuscles  collect  in  the  bottom  of  the 
graduated  tube.  A  measured  volume  of  blood  is  centrifuged 
until  no  further  shrinkage  in  volume  of  the  corpuscles  can  be 
detected  in  the  hsematocrit.  The  same  volume  of  blood  is  then 
added  to  each  of  a  series  of  solutions  whose  concentration  dimin- 
ishes progressively,  and  the  volume  of  the  corpuscles  is  determined 
as  in  pure  blood.  In  this  way  the  concentration  of  the  solution 
is  found,  in  which  the  volume  of  the  corpuscles  is  the  same  as  in 
the  undiluted  blood.  By  proceeding  in  a  similar  manner  with 
solutions  of  different  substances,  a  series  of  isotonic  coefficients 
can  be  determined.  The  following  table  gives  a  comparison  of 
the  isotonic  coefficients  of  various  substances  obtained  by  the 
plasmolytic,  blood  corpuscle  and  hsematocrit  methods.  The  iso- 
tonic coefficients  are  referred  to  that  of  cane  sugar  as  unity. 

There  are  other  methods  which  may  be  used  for  the  comparison 
of  osmotic  pressures,  among  which  may  be  mentioned  that  due  to 
Wladimiroff,f  involving  the  use  of  bacteria,  and  the  interesting 

*  Ibid.,  17,  164  (1895). 
*  t  Zeit.  phys.  Chem.,  7,  529  (1891). 


206 


THEORETICAL  CHEMISTRY 


Substance. 

Plasmolytic 
Method. 

Corpuscle 
Method. 

Hsematocrit 
Method. 

1.00 

1.00 

1.00 

MgSO4  

1.09 

1.27 

1.10 

KNO3 

1   67 

1  74 

1   84 

NaCl 

1.69 

1.75 

1  74 

CHs.COOK 

1.67 

1.66 

1.67 

CaCl2  

2.40 

2.36 

2.33 

method  developed  by  Tammann,*  in  which  artificially  prepared 
membranes  are  employed. 

Osmotic  Pressure  and  Diffusion.  That  there  is  a  very  close 
connection  between  osmotic  pressure  and  diffusion,  has  already 
been  pointed  out.  In  fact  the  osmotic  pressure  of  a  solution  may 
be  regarded  as  the  driving  force  which  causes  the  molecules  of  a 
dissolved  substance  to  distribute  themselves  uniformly  through- 
out the  solution. 

The  process  of  diffusion  was  first  systematically  investigated 
by  Graham  f  in  1850,  but  it  was  not  until  five  years  later  that 
the  general  law  of  diffusion  was  enunciated  by  Fick.|  He  proved 
theoretically  and  experimentally  that  the  quantity  of  solute,  ds, 
which  diffuses  through  an  area  A,  in  a  time  dt,  when  the  concen- 
tration changes  by  an  amount  dc,  in  a  distance  dx,  at  right 
angles  to  the  plane  of  A,  is  given  by  the  equation 

ds=  -DA^-dt, 
dx 

in  which  D  is  a  constant,  known  as  the  coefficient  of  diffusion. 
Interpreting  the  equation  of  Fick  in  words,  we  see  that  the  coeffi- 
cient, of  diffusion  is  the  amount  of  solute  which  will  cross  1  square 
centimeter  in  1  second,  if  the  change  of  concentration  per  centi- 
meter is  unity. 

The  phenomena  of  diffusion  have  also  been  investigated  by 

*  Wied.  Ann.,  34,  299  (1888). 

t  Phil.  Trans.  (1850),  p.  1,  805;   (1851),  p.  483. 

J  Pogg.  Ann.,  94,  59  (1855). 


DILUTE  SOLUTIONS  AND   OSMOTIC  PRESSURE          207 

Nernst  *  and  Planck,  f  If  we  have  a  tall  cylindrical  vessel  con- 
taining a  solution  of  a  non-electrolyte  in  its  lower  part,  and  pure 
water  at  the  top,  the  solute  will  slowly  diffuse  upward  into  the 
water. 

Assuming  the  osmotic  pressure  at  a  height  x,  to  be  P,  and  letting 
A  denote  the  area  of  cross-section  of  the  cylinder,  the  solute  in 
the  layer  whose  volume  is  A  dx,  will  be  subjected  to  a  force  equal 
to  —  A  dP,  the  negative  sign  indicating  that  the  force  acts  in  the 
direction  of  diminishing  pressure.  If  c  is  the  concentration  in 
mols  per  cubic  centimeter,  the  force  acting  on  each  molecule  in 
this  layer  will  be 

-  A.m  =  __  l.^L 

cA    dx  c    dx 

Let  F  denote  the  force  necessary  to  drive  a  single  molecule  through 
the  solution  with  the  velocity  of  one  centimeter  per  second.  Since 
the  velocity  is  constant,  the  resistance  due  to  the  viscosity  of  the 
medium  must  also  be  denoted  by  F.  The  velocity  attained  will 
be 

JL  &P_ 

cF '  dx ' 

If  dN  represents  the  number  of  molecules  crossing  each  layer  in 
a  time  dt,  then,  since  the  number  crossing  unit  area  per  second 
is  proportional  to  the  concentration  and  to  the  mean  velocity  of 
the  molecules,  we  shall  have 

dN  =  -  ~-~Acdty 
cF   dx 

or,. 

,A7  1   ,  dP  ,. 

dN=     ~FATxdL 

When  the  solution  is  dilute,  we  may  apply  the  general  equation, 
PV  =  RT,  remembering  that  V  =  1/c.  Substituting  in  the  pre- 
ceding equation,  we  have 

,A,  RT     dc  .. 

dN  =  -    -=r  A  -j-  dt. 
F       dx 

*  Zeit.  phys.  Chem.,  2,  40,  615  (1888);  4,129  (1889). 
t  Wied.  Ann.,  40,  561  (1890). 


208  THEORETICAL  CHEMISTRY 

Comparing  this  equation  with  that  of  Fick,  we  see  that  the 
coefficient  of  diffusion  D,  corresponds  to  the  factor,  RT/F.  From 
the  equation  of  Nernst  it  is  possible  to  calculate  the  force  required 
to  drive  a  molecule  of  solute  through  the  solution  with  unit  veloc- 
ity. Thus,  solving  the  above  equation  for  F,  we  have 


A        , 

F  =  -  —A-j-dt. 
dN     dx 

By  means  of  this  equation,  it  has  been  calculated  that  the  force 
necessary  to  drive  one  molecule  of  formic  acid  through  water 
with  a  velocity  of  one  centimeter  per  second  at  0°  C.  is  equal  to 
the  weight  of  4,340,000,000  kilograms.  It  is  difficult  at  first  to 
realize  that  such  enormous  forces  are  operative  in  solutions,  but 
when  one  considers  the  minute  size  of  the  molecules  and  the  great 
resistance  offered  by  the  medium,  it  becomes  evident  that  a  very 
large  driving  force  must  be  applied  to  produce  an  appreciable 
movement  of  the  solute  through  the  solution. 

Principle  of  Soret.  If  a  solution  is  maintained  at  a  uniform 
temperature  it  will  ultimately  become  homogeneous;  if,  on  the 
other  hand,  two  parts  of  a  homogeneous  solution  are  kept  at 
different  temperatures  for  some  time,  the  solution  will  become 
more  concentrated  in  the  colder  portion.  This  phenomenon  was 
first  investigated  by  Soret.*  The  experiments  of  Soret  are  of 
special  interest,  since  they  furnish  a  means  of  determining  the 
influence  of  temperature  on  osmotic  pressure.  Thus,  if  the  law 
of  Gay-Lussac  holds  for  osmotic  pressure,  the  colder  portion  of  a 
solution  should  increase  in  concentration  by  1/273  for  each  degree 
of  difference  in  temperature.  The  experimental  results  are  in 
satisfactory  agreement  with  the  requirements  of  theory,  and  con- 
stitute another  proof  of  the  applicability  of  the  gas  laws  to  dilute 
solutions. 

Lowering  of  Vapor  Pressure.  It  has  long  been  known  that 
the  vapor  pressure  of  a  solution  is  less  than  that  of  the  pure  sol- 
vent, provided  the  solute  is  non-volatile.  The  investigations  of 
von  Babo  and  Wiillner  f  on  the  lowering  of  vapor  pressure  of 

*  Ann.  Chem.  Phys.  (5),  22,  293  (1881). 
t  Pogg.  Ann.,  103,  529  (1858). 


DILUTE  SOLUTIONS  AND   OSMOTIC   PRESSURE 


209 


various  liquids  when  non-volatile  substances  are  dissolved  in 
them,  resulted  in  the  following  generalizations: —  (1)  The  lower- 
ing  of  the  vapor  pressure  of  a  solution  is  proportional  to  the  amount 
of  solute  present ;  and  (2)  For  the  same  solution,  the  lowering  of  the 
vapor  pressure  of  the  solvent  by  a  non-volatile  solute  is  at  all  tempera- 
tures a  constant  fraction  of  the  vapor  pressure  of  the  pure  solvent. 

In  1887,  Raoult,*  as  the  result  of  an  exhaustive  experimental 
investigation,  enunciated  the  following  laws: — (1)  When  equi- 
molecular  quantities  of  different  non-volatile  solutes  are  dissolved  in 
equal  volumes  of  the  same  solvent,  the  vapor  pressure  of  the  solvent  is 
lowered  by  a  constant  amount]  and  (2)  The  ratio  of  the  observed 
lowering  of  the  vapor  pressure  to  the  vapor  pressure  of  the  pure  sol- 
vent is  equal  to  the  ratio  of  the  number  of  mols  of  solute  to  the  total 
number  of  mols  in  the  solution.  The  ratio  of  the  observed  lowering 
to  the  original  vapor  pressure  is  called  the  relative  lowering  of  the 
vapor  pressure.  Letting  p\  amd  pz  denote  the  vapor  pressures  of 
solvent  and  solution,  Raoult's  second  law  may  be  put  in  the 
form 

Pi-  P2  =      n 
pi          N  +  n' 

in  which  n  and  N  represent  the  number  of  mols  of  solute  and 
solvent  respectively.  Some  of  Raoult's  results  for  ethereal  solu- 
tions are  given  in  the  accompanying  table. 


Mols  of  Solute 

PI  —  Pt 
PI 

Substance. 

per  100  mols 
of  Solution. 

for  Solution. 

for  1  molar 
per  cent 
Solution. 

Turpentine  

8  95 

0  0885 

0  0099 

Methyl  salicylic  acid  

2  91 

0  026 

0  0089 

Methyl  benzoic  acid  
Benzoic  acid 

9.60 
7  175 

0.091 
0  070 

0.0095 
0  0097 

Trichloracetic  acid 

11  41 

0  120 

0  0105 

Aniline    . 

7  66 

0  081 

0  0106 

The  results  given  in  the  fourth  column  of  the  table  are  nearly 

*  Compt.  rend.,  104,  1430  (1887);  Zeit.  phys.  Chem.,  2,  372  (1888);  Ann. 
Chem.  Phys.  (6),  15,  375  (1888). 


210  THEORETICAL  CHEMISTRY 

constant,  and  are  in  close  agreement  with  the  theoretical  value 
of  the  relative  lowering  of  a  1  molar  per  cent  solution  calculated 
as  follows:  — 

^—  -  =  Tnff4T7  =  °-0099- 
pi 

When  the  solution  is  very  dilute,  the  number  of  mols  of  solute 
is  negligible  in  comparison  with  the  number  of  mols  of  solvent, 
and  the  equation  of  Raoult  may  be  written 

Pi  -  P2  =  _rc 

Pi          N' 

Since  n  =  g/m,  and  N  =  W/M,  where  g  and  W  are  the  weights  of 
solute  and  solvent  respectively,  and  m  and  M  are  the  correspond- 
ing molecular  weights,  the  above  equation  becomes 

Pi-  P2  _  gM 
pi          Wm' 

This  equation  enables  us  to  calculate  the  molecular  weight  of  a 
dissolved  substance  from  the  relative  lowering  of  the  vapor 
pressure  produced  by  the  solution  of  a  known  weight  of  solute  in 
a  known  weight  of  solvent.  Solving  the  equation  for  m,  we 

have 

Pl 


— 

W      PI-  Pz 

As  an  illustration  of  the  application  of  this  equation,  we  may 
take  the  determination  of  the  molecular  weight  of  ethyl  benzoate 
from  the  following  experimental  data:  —  The  vapor  pressure  at 
80°  C.  of  a  solution  of  2.47  grams  of  ethyl  benzoate  in  100  grams 
of  benzene  is  742.6  mm.:  the  vapor  pressure  of  pure  benzene  at 
80°  C.  is  751.86  mm.  Substituting  in  the  equation,  we  have 

=  2.47  X  78  751.86 

100       '  751.86  -  742.6  ~ 

The  molecular  weight  calculated  from  the  formula,  C6H5COO.C2H5, 
is  150. 

The  difficulties  which  attend  the  accurate  measurement  of  the 
vapor  pressure  of  a  solution  by  the  static  method  have  already 


DILUTE  SOLUTIONS  AND  OSMOTIC  PRESSURE          211 


been  mentioned.  While  there  are  other  methods  which  are  pref- 
erable for  the  determination  of  the  molecular  weight  of  dissolved 
substances,  the  vapor  pressure  method  has  one  marked  advan- 
tage, —  it  can  be  used  for  the  same  solution  at  widely  divergent 
temperatures.  The  method  devised  by  Walker  and  already 
described  in  connection  with  the  determination  of  the  vapor 
pressure  of  pure  liquids  (p.  92)  is  well  adapted  to  the  measure- 
ment of  the  vapor  pressure  of  solutions. 

Connection  between  Lowering  of  Vapor  Pressure  and  Osmotic 
Pressure.     The    relation    between    osmotic    pressure    and    the 
lowering  of  vapor  pressure  has  been  derived 
in   the    following    manner   by   Arrhenius.* 
Imagine  a  very  dilute  solution  contained  in 
the  wide  glass  tube  A,  Fig.  64.     The  tube, 

A,  is  closed  at  its  lower  end  with  a  semi- 
permeable  membrane,  and  dips  into  a  vessel, 

B,  which  contains  the  pure  solvent.     The 
entire  apparatus  is  covered  by  a  bell-jar  C, 
and   the  enclosed    space    exhausted.     Let 
h    be   the  difference  in  level  between  the 
solvent  and   solution   when  equilibrium  is 
established,   that   is,  when  the  hydrostatic 
pressure  of  the  column  of  liquid  is  equal  to 
the  osmotic  pressure.     When  equilibrium  is    _ 
attained,  the  vapor  pressure  of  the  solution 

at  .h  will  be  equal  to  the  vapor  pressure  of 
the  solvent  at  this  height.  If  the  vapor  pressure  of  the  pure  sol- 
vent is  pi,  and  p2  is  the  vapor  pressure  of  the  solution  at  h,  we 
shall  have 

Pi  —  Pz  =  hd,  (1) 

where  d  denotes  the  density  of  the  vapor.  Let  v  be  the  volume  of 
1  mol  of  solvent  in  the  state  of  vapor,  then 

Piv  =  RT, 
and 

RT 

v  — 


f 


T^J^^-- 


Fig.  64. 


Zeit.  phys.  Chem.,  3,  115  (1889). 


212  THEORETICAL  CHEMISTRY 

If  the  molecular  weight  of  the  solvent  is  M,  we  may  replace  v 
by  M/d,  when  the  preceding  equation  becomes 


pi 


The  solution  being  very  dilute  the  osmotic  pressure  may  be  cal- 
culated from  the  equation 

PV  =  nRT, 

where  P  is  the  osmotic  pressure  of  the  solution,  V  the  volume 
of  the  solution  containing  1  mol  of  solute,  and  n  the  number  of 
mols  of  solute  present.  If  s  represents  the  density  of  the  solvent 
and  also  of  the  solution,  since  it  is  very  dilute,  we  may  write 

P  =  hs, 
and 

F-2, 

S 

where  g  is  the  number  of  grams  of  the  solvent  in  which  the  n 
mols  of  solute  are  dissolved.  Substituting  these  values  of  P  and 
V  in  the  general  equation,  we  have 

PV  =  nRT  =  hg, 
and  solving  for  h, 

.-•-?. 

Substituting  the  values  of  d  and  h,  given  hi  equations  (2)  and 
(3),  in  equation  (1)  we  have 


Rearranging  equation  (4),  and  remembering  that  N  =  g/M,  we 
obtain 

Pi  "ft  _  n 

Pi      ~N' 

This  equation  it  will  be  seen,  is  identical  with  that  derived  experi- 
mentally by  Raoult  for  very  dilute  solutions. 


DILUTE  SOLUTIONS  AND  OSMOTIC  PRESSURE         213 

Van't  Hoff  showed,  by  an  application  of  thermodynamics  to 
dilute  solutions,  that  the  relation  between  osmotic  pressure  and 
the  relative  lowering  of  the  vapor  pressure  is  expressed  by  the 
equation 

Pi  -  Pz  =  MP 
pi          sRT' 

in  which  the  symbols  have  the  same  significance  as  above.     This 
equation  may  be  reconciled  easily  with  the  equation  of  Raoult. 
If  n  in  equation  (5)  be  replaced   by  its  equal,  PV/RT,  the 
equation  becomes 

Pi  -  pa 


pi          NET' 

But  V  =  NM/s,  hence 

Pi  -  Pz  =  MPf 
pi          sRT* 

This  equation  shows  that  the  relative  lowering  of  the  vapor  pressure 
is  directly  proportional  to  the  osmotic  pressure. 

Elevation  of  the  Boiling-Point.  Just  as  the  vapor  pressure  of 
a  solution  is  less  than  that  of  the  pure  solvent,  so  the  boiling-point 
of  a  solution  is  correspondingly  higher  than  the  boiling-point  of 
the  solvent.  It  follows  that  when  equimolecular  quantities  of 
different  substances  are  dissolved  hi  equal  volumes  of  the  same 
solvent,  the  elevation  of  the  boiling-point  is  constant.  Thus,  the 
molecular  weight  of  any  soluble  substance  may  be  determined  by 
comparing  its  effect  on  the  boiling-point  of  a  particular  solvent, 
with  that  of  a  solute  of  known  molecular  weight.  The  elevation 
in  boiling-point  produced  by  dissolving  1  mol  of  a  solute  in  100 
grams,  or  100  cubic  centimeters,  of  a  solvent  is  termed  the  moke- 
ular  elevation,  or  boiling-point  constant  of  the  solvent.  In  deter- 
mining the  boiling-point  constant  of  a  solvent,  a  fairly  dilute 
solution  is  employed  and  the  elevation  in  the  boiling-point  is 
observed;  the  value  of  the  constant  is  then  calculated  on  the 
.assumption  that  the  elevation  in  boiling-point  is  proportional  to 
the  concentration. 

If  g  grams  of  a  substance  of  unknown  molecular  weight  m, 
are  dissolved  in  W  grams  of  solvent,  and  the  boiling-point  is  raised 


214 


THEORETICAL  CHEMISTRY 


A  degrees,  then,  since  m  grams  of  the  substance  when  dissolved 
in  100  grams  of  solvent,  produce  an  elevation  of  K  degrees  (the 
molecular  elevation),  it  follows  that 

IQOg.A.       .K 
W 

therefore, 

Q 


The  accompanying  table  gives  the  boiling-point  constants  for 
100  grams  and  100  cubic  centimeters  of  some  of  the  more  com- 
mon solvents. 


Solvent. 

Molecular  Elevation. 

100  gr. 

100  cc. 

Water  

5.2 

11.5 
21.0 
16.7 
26.7 
35.6 
30.1 

5.4 
15.6 
30.3 
22.2 
32.8 

Ethyl  alcohol  

Ether  

Acetone 

Benzene 

Chloroform 

Pyridine  

As  an  example  of  the  calculation  of  the  molecular  weight  of  a 
dissolved  substance  by  the  above  formula,  we  may  take  the 
calculation  of  the  molecular  weight  of  camphor  in  acetone  from 
the  following  data :  — 

When  0.674  gram  of  camphor  is  dissolved  in  6.81  grams  of 
acetone,  the  boiling-point  of  the  solvent  is  raised  1°.09.  Substitut- 
ing in  the  formula,  we  have 

0.674 


m  =  100  X  16.7  X 


6.81  X  1.09 


=  151. 


The  molecular  weight   of   camphor   according   to   the   formula 
Cio  H160,  is  152. 

The  molecular  elevation  of  the  boiling-point  can  be  calculated 
by  means  of  the  formula, 

0.02  T2 


K  = 


w 


DILUTE  SOLUTIONS  AND  OSMOTIC  PRESSURE 


215 


in  which  T  is  the  absolute  boiling-point  of  the  solvent,  and  w  is 
the  heat  of  vaporization  for  1  gram  of  the  solvent  at  its  boiling- 
point.  This  formula  will  be  derived  in  a  subsequent  paragraph 
of  this  chapter.  The  calculated  values 
of  K  are  in  close  agreement  with  the 
values  obtained  experimentally  by  Raoult 
and  others.  As  an  example,  the  calcu- 
lated value  of  the  molecular  elevation 
for  water,  the  heat  of  vaporization  of 
which  at  100°  C.  is  537  calories,  is 


K 


0.02  X  (373)2 
537 


=  5.2, 


a  value  in  exact  agreement  with  the  exper- 
imental value  given  in  the  table. 

Experimental  Determination  of  Mo- 
lecular Weight  by  the  Boiling-Point 
Method.  One  of  the  simplest  and  most 
convenient  of  the  various  forms  of  appa- 
ratus which  have  been  devised  for  the 
determination  of  the  boiling-point  of 
solutions,  is  that  developed  by  Jones,* 
and  shown  hi  Fig.  65.  The  liquid  whose 
boiling-point  is  to  be  determined  is  intro- 
duced into  the  vessel  A,  which  already 
contains  a  platinum  cylinder  P,  em- 
bedded in  some  glass  beads.  Sufficient 
liquid  is  introduced  to  insure  the  com- 
plete covering  of  the  bulb  of  the  ther- 
mometer, as  shown  in  the  sketch. 


Fig.  65. 


The  side  tube  of  A  is  connected  with  a  condenser,  C.  The 
vessel  A,  is  surrounded  by  a  thick  jacket  of  asbestos  J,  and  rests 
on  a  piece  of  asbestos  board  in  which  a  circular  hole  is  cut,  and 
over  which  a  piece  of  wire  gauze  is  placed.  The  liquid  is  heated 
by  means  of  a  burner,  B.  The  platinum  cylinder  is  the  feature 
which  differentiates  this  apparatus  from  the  various  other  forms 
*  Am.  Chem.  Jour.,  19,  581  (1897). 


216  THEORETICAL  CHEMISTRY 

of  boiling-point  apparatus.  It  has  the  two-fold  object  of  pre- 
venting the  condensed  solvent  from  coming  in  direct  contact  with 
the  bulb  of  the  thermometer,  and  of  reducing  the  effect  of  radia- 
tion to  a  minimum.  The  liquid  in  A  is  boiled,  using  a  very  small 
flame,  until  the  thermometer  remains  constant;  this  temperature 
is  taken  as  the  boiling-point  of  the  liquid.  The  apparatus  is  now 
emptied  and  dried.  A  weighed  amount  of  the  liquid  is  then  intro- 
duced into  A,  and  to  this  is  added  a  known  weight  of  solute;  the 
thermometer  is  replaced  and  the  boiling-point  of  the  solution  is 
determined.  The  difference  between  the  readings  of  the  thermom- 
eter when  immersed  in  the  solution,  and  in  the  solvent  alone,  gives 
the  boiling  point  elevation.  For  further  details  concerning  the 
boiling-point  method  as  applied  to  the  determination  of  molecular 
weights,  the  student  is  referred  to  any  one  of  the  standard  labo- 
ratory manuals. 

Osmotic  Pressure  and  Boiling-Point  Elevation.  Imagine  a 
dilute  solution  containing  n  mols  of  solute  in  G  grams  of  solvent, 
and  let  dT  be  the  elevation  in  the  boiling-point.  Suppose  a  large 
quantity  of  the  solution  to  be  introduced  into  a  cylinder,  fitted 
with  a  frictionless  piston,  and  closed  at  the  bottom  by  a  semi- 
permeable  membrane.  Let  the  cylinder  and  contents  be  raised 
to  the  absolute  temperature  T°,  the  boiling-point  of  the  solvent, 
and  then  let  pressure  be  exerted  on  the  piston  just  sufficient 
to  overcome  the  osmotic  pressure  of  the  solution.  In  this  way,  let 
a  quantity  of  solvent  corresponding  to  1  mol  of  solute  be  forced 
through  the  semi-permeable  membrane.  The  volume  V,  thus 
expelled  is  the  volume  corresponding  to  G/n  grams  of  solvent. 
If  the  osmotic  pressure  of  the  solution  is  P,  then  the  work  done  in 
moving  the  piston  and  expelling  the  solvent  is  PV.  Now  let  the 
portion  of  the  solvent  which  has  been  forced  through  the  semi- 
permeable  membrane  be  vaporized.  For  this  operation  G/n  .  w 
calories  will  be  required,  w  being  the  heat  of  vaporization  for  1 
gram  of  solvent  at  its  boiling-point.  Then  let  the  entire  system 
be  raised  to  the  boiling-point  of  the  solution  (T  +  dT)°,  the  pre- 
viously expelled  G/n  grams  of  vapor  being  allowed  to  mix  with 
the  solution.  The  heat  of  vaporization  lost  at  T°  is  thus  recov- 
ered at  the  slightly  higher  temperature,  (T  +  dT)°.  Finally,  the 


DILUTE  SOLUTIONS  AND  OSMOTIC   PRESSURE          217 

entire  system  is  cooled  to  T°,  and  is  thus  restored  to  its  original 
state.  Applying  the  well-known  thermodynamic  relation,  that 
the  ratio  of  the  work  done  to  the  heat  absorbed,  is  the  same  as 
the  ratio  of  the  difference  in  temperature  to  the  absolute  initial 
temperature  of  the  system,  we  have 

PV       dT 

~       =  "T' 

—  •  w 
n 

therefore, 

«-¥•*•        « 

But,  since  PV  =  RT,  equation  (1)  may  be  written 


.. 

w      G 

If  n  =  1  and  G  =  100  grams,  then  dT  =  K,  (the  molecular  eleva- 
tion of  the  boiling-point),  or 

RT* 

100  w' 

Or  putting  R  =  2  calories,  we  have 

X-*S*H.  (2) 

W 

Equation  (1)  shows  that  the  osmotic  pressure  of  a  solution  is  directly 
proportional  to  the  elevation  of  the  boiling-point.  Equation  (2)  was 
originally  derived  by  van't  Hoff  at  about  the  time  when  Raoult 
determined  the  values  of  K  experimentally. 

Lowering  of  the  Freezing-Point.  Of  all  the  methods  employed 
for  the  determination  of  molecular  weights  in  solution,  the  freez- 
ing-point method  is  the  most  accurate  and  the  most  widely  used. 
It  was  pointed  out  by  Blagden  *  over  a  century  ago,  that  the  de- 
pression of  the  freezing-point  of  a  solvent  by  a  dissolved  substance  is 
directly  proportional  to  the  concentration  of  the  solution.  When 
equimolecular  quantities  of  different  substances  are  dissolved  in 
equal  volumes  of  the  same  solvent,  the  lowering  of  the  freezing- 
point  is  constant.  The  molecular  weight  of  any  soluble  sub- 

*  Phil.  Trans.,  78,  277  (1788). 


218 


THEORETICAL  CHEMISTRY 


stance  can  be  found,  as  in  the  boiling-point  method,  by  comparing 
its  effect  on  the  freezing-point  of  a  solvent  with  that  of  a  solute 
of  known  molecular  weight.  The  molecular  lowering  of  the  freezing- 
point,  or  the  freezing-point  constant,  of  a  solvent  is  defined  as  the  de- 
pression of  the  freezing-point  produced  by  dissolving  1  mol  of  solute 
in  100  grams  or  100  cubic  centimeters  of  solvent.  The  freezing-point 
constants  of  a  few  common  solvents  are  given  in  the  following  table. 


Molecular  I 

)epression. 

100  gr. 

100  cc. 

Water                                               

18.5 

18.5 

Benzene  .  .             .  .       .    .           x   

50 

56 

Phenol  

74 

Naphthalene  

69 

Acetic  acid 

39 

41 

Van't  Hoff  showed  that  the  molecular  lowering  of  the  freezing- 
point  of  a  solvent  K,  can  be  calculated  from  the  absolute  freez- 
ing-point T,  and  the  heat  of  fusion  w,  for  1  gram  of  solvent  at 
the  temperature  T,  by  means  of  the  formula 

K  _  0.02  T2 

w 

This  expression  is  analogous  to  that  which  applies  to  the  molec- 
ular elevation  of  the  boiling-point.  The  agreement  between  the 
observed  and  the  calculated  values  of  K  is  very  satisfactory,  as 
the  following  calculation  for  water  shows :  — 

0.02) 


oU 

It  is  of  interest  to  note  that  the  calculated  value  of  K  for  water  is 
lower  than  the  experimental  values  originally  obtained  by  Raoult 
and  others.  Subsequent  experiments,  carried  out  with  greater 
care  and  better  apparatus,  by  Raoult,  Abegg  and  Loomis  gave 
values  in  close  agreement  with  that  derived  theoretically.  A 
formula  analogous  to  that  employed  for  the  calculation  of  the 
molecular  weight  of  a  dissolved  substance  from  the  elevation  it 


DILUTE  SOLUTIONS  AND  OSMOTIC   PRESSURE          219 

produces  in  the  boiling-point  of  a  solvent,  may  be  used  for  the 
calculation  of  molecular  weight  from  freezing-point  depression. 
Thus,  if  g  grams  of  solute  when  dissolved  in  W  grams  of  solvent 
produce  a  depression  A  of  the  freezing-point  of  the  solvent,  the 
molecular  weight  m,  is  given  by  the  formula 


where  K  is  the  molecular  lowering  of  the  freezing-point. 

EXAMPLE.  When  1.458  grams  of  acetone  are  dissolved  in 
100  grams  of  benzene,  the  freezing-point  of  the  solvent  is  de- 
pressed 1.22°,  therefore  the  molecular  weight  of  acetone  is 

1 


The  molecular  weight  of  acetone,  calculated  from  the  formula 
C3H6O,  is  58. 

In  order  to  obtain  trustworthy  results  with  the  freezing-point 
method,  it  is  necessary  that  only  the  pure  solvent  separate  out 
when  the  solution  freezes,  and  that  excessive  overcooling  be 
avoided.  When  too  great  overcooling  occurs,  the  subsequent 
freezing  of  the  solution  results  in  the  separation  of  so  large  an 
amount  of  solvent  in  the  solid  state,  that  the  observed  freezing 
point  corresponds  to  the  equilibrium  temperature  of  a  more 
concentrated  solution  than  that  originally  prepared.  A  formula 
for  the  correction  of  the  concentration,  due  to  excessive  overcooling, 
has  been  derived  by  Jones.*  If  the  overcooling  of  the  solution 
in  degrees  be  represented  by  u,  the  heat  of  fusion  of  1  gram  of  sol- 
vent at  the  freezing-point  by  w,  and  the  specific  heat  of  the  sol- 
vent by  c,  then  the  fraction  of  the  solvent  which  will  solidify,  /, 
may  be  calculated  by  the  formula, 

.  _  cu 
w' 

When  water  is  used  as  the  solvent,  c  =  1  and  w  =  80.  There- 
fore, for  every  degree  of  overcooling,  the  fraction  of  the  solvent 
separating  as  ice  will  be  1/80,  and  the  concentration  of  the  original 

*  Zeit.  phys.  Chem.,  12,  624  (1893). 


220 


THEORETICAL  CHEMISTRY 


solution  is  increased  by  just  so  much.  It  is  simpler,  however,  to 
apply  the  correction  directly  to  the  freezing-point  depression  in- 
stead of  to  the  concentration. 

Experimental  Determination  of  Molecular  Weight  by  the 
Freezing-Point  Method.  The  apparatus  almost  universally  em- 
ployed for  the  determination  of  molecular  weights  by  the  freezing- 
point  method  is  that  devised  by  Beckmann,* 
and  shown  in  Fig.  66.  It  consists  of  a  thick- 
walled  test  tube  A,  provided  with  a  side  tube, 
and  fitted  into  a  wider  tube  AI,  thus  surround- 
ing A  with  an  air  space. 

The  whole  is  fitted  into  the  metal  cover  of 
a  large  battery  jar,  which  is  filled  with  a  freez- 
ing mixture  whose  temperature  is  several  de- 
grees below  the  freezing-point  of  the  solvent. 

The  tube  A  is  closed  by  a  cork  stopper, 
through  which  passes  the  thermometer  and 
stirrer.  The  thermometer  is  generally  of  the 
Beckmann  differential  type.  This  instrument 
has  a  scale  about  6°  in  length,  each  degree 
being  divided  into  hundredths;  the  quantity 
of  mercury  in  the  bulb  can  be  varied  by  means 
of  a  small  reservoir  at  the  top  of  the  scale,  so 
that  the  zero  of  the  instrument  can  be  adjusted 
for  use  with  solvents  of  widely  different  freez- 
ing-points. In  carrying  out  a  determination 
with  the  Beckmann  apparatus,  a  weighed 
quantity  of  solvent  is  placed  in  A,  and  the  temperature  of  the 
refrigerating  mixture  regulated  so  as  to  be  not  more  than  5°  below 
the  freezing-point  of  the  solvent.  The  tube  A  is  removed  from  its 
jacket,  and  is  immersed  in  the  freezing  mixture  until  the  solvent 
begins  to  freeze.  It  is  then  replaced  in  the  jacket  AI,  and  the 
solvent  is  vigorously  stirred.  The  thermometer  rises  during  the 
stirring  until  the  true  freezing-point  is  reached,  after  which  it 
remains  constant.  This  temperature  is  taken  as  the  freezing 
temperature  of  the  solvent. 

*  Zeit.  phys.  Chem.,  2,  683  (1888) . 


Fig.  66. 


DILUTE  SOLUTIONS  AND  OSMOTIC  PRESSURE          221 

The  tube  A  is  now  removed  from  the  freezing  mixture,  and  a 
weighed  amount  of  the  substance  whose  molecular  weight  is  to 
be  determined  is  introduced.  When  the  substance  has  dissolved, 
the  tube  is  replaced  in  AI  and  the  solution  cooled  not  more  than 
a  degree  below  its  freezing-point.  A  small  fragment  of  the  solid 
solvent  is  dropped  into  the  solution,  which  is  then  stirred 
vigorously  until  the  thermometer  remains  constant.  The  maxi- 
mum, temperature  is  taken  as  the  freezing-point  of  the  solution. 
The  difference  between  the  freezing-points  of  solution  and 
solvent  is  the  depression  sought.  For  further  details  con- 
cerning the  determination  of  molecular  weights  by  the  freezing- 
point  method,  the  student  is  referred  to  a  physico-chemical 
laboratory  manual. 

Osmotic  Pressure  and  Freezing-Point  Depression.  Let  dT 
be  the  freezing-point  depression  produced  by  n  mols  of  solute  in 
G  grams  of  solvent,  the  solution  being  dilute.  Imagine  a  large 
quantity  of  this  solution  to  be  confined  within  a  cylinder  fitted 
with  a  frictionless  piston,  the  bottom  of  the  cylinder  being  closed 
by  a  semi-permeable  membrane.  Let  the  cylinder  and  contents 
be  cooled  to  the  freezing  temperature  of  the  solvent  T,  and  then 
let  pressure  be  applied  to  the  piston  until  a  quantity  of  solvent 
corresponding  to  1  mol  of  solute  is  forced  through  the  semi-perme- 
able membrane.  This  requires  an  expenditure  of  energy  equiva- 
lent to  PV,  where  P  is  the  osmotic  pressure  of  the  solution  and  V 
is  the  volume  of  solvent  expelled.  The  volume  V  is  clearly  the 
volume  of  G/n  grams  of  solvent.  Now  let  the  expelled  portion 

C1 

of  solvent  be  frozen  and  the  system  deprived  of  —  w  calories  of 

heat,  where  w  is  the  heat  of  fusion  of  1  gram  of  the  solvent  at  the 
temperature  T. 

The  temperature  of  the  solution  is  then  lowered  to  its  freezing- 
point  (T  —  dT),  and  the  G/n  grams  of  solidified  solvent  dropped 
into  it.  The  solidified  solvent  melts,  thereby  restoring  to  the 
system  at  the  temperature  (T  —  dT),  the  heat  of  fusion  formerly 
taken  from  it.  Finally,  the  temperature  of  the  system  is  raised 
to  T,  the  initial  temperature  of  the  cycle.  Applying  the  familiar 
thermodynamic  relation,  that  the  ratio  of  the  work  done  to  the 


222  THEORETICAL  CHEMISTRY 

heat  absorbed,  is  the  same  as  the  ratio  of  the  difference  in  temper- 
ature to  the  initial  absolute  temperature,  we  have 

PV      AT 


rn 

—  w 
n 

From  which  we  obtain 


w      G 
But  PV  =  RT,  hence  equation  (1)  becomes 


w     G 

If  n  =  1  and  G  =  100  grams,  then  dT  =  K,  the  molecular  lower- 
ing of  the  freezing-point,  and 

RT* 
~  100  W' 

Or  putting  R  =  2  calories,  we  have 


An  equation  to  which  reference  has  already  been  made. 

It  is  evident  from  equation  (1)  that  the  osmotic  pressure  of  a 
solution  is  directly  proportional  to  the  freezing  point  depression. 

Molecular  Weight  in  Solution.  As  has  been  pointed  out,  the 
molecular  weight  of  a  dissolved  substance  can  be  readily  calcu- 
lated, provided  that  the  osmotic  pressure  of  a  dilute  solution  of 
known  concentration  at  known  temperature  is  determined.  But 
the  experimental  difficulties  attending  the  direct  measurement 
of  osmotic  pressure  are  so  great,  that  it  is  customary  to  employ 
other  methods  based  upon  properties  of  dilute  solutions  which 
are  proportional  to  osmotic  pressure.  We  have  shown  that  in 
dilute  solutions  osmotic  pressure  is  directly  proportional  (1)  to 
the  relative  lowering  of  the  vapor  pressure,  (2)  to  the  elevation 
of  the  boiling-point,  and  (3)  to  the  depression  of  the  freezing-point. 

From  this  it  follows,  that  equimolecular  quantities  of  different 
substances  dissolved  in  equal  volumes  of  the  same  solvent,  exert  the 


DILUTE  SOLUTIONS  AND  OSMOTIC   PRESSURE          223 

same  osmotic  pressure,  and  produce  the  same  relative  lowering  of 
vapor  pressure,  the  same  elevation  of  boiling-point,  and  the  same 
depression  'of  freezing-point.  Since  equimolecular  quantities  of 
different  substances  contain  the  same  number  of  molecules,  it  is 
evident  that  the  magnitude  of  osmotic  pressure,  relative  lowering  of 
vapor  pressure,  elevation  of  boiling-point  and  depression  of  freezing- 
point,  is  dependent  upon  the  number  of  particles  present  in  the  solu- 
tion and  is  independent  of  their  nature.  It  has  been  pointed  out 
by  Nernst  that  any  process  which  involves  the  separation  of 
solvent  from  solute,  may  be  employed  for  the  determination  of 
molecular  weights.  A  little  reflection  will  convince  the  reader 
that  the  four  methods  discussed  in  this  chapter  involve  such  separ- 
ation. Both  van't  Hoff  and  Raoult  emphasized  the  fact  that  the 
formulas  derived  for  the  determination  of  molecular  weights  in  so- 
lution depend  upon  assumptions  which  are  valid  only  for  dilute 
solutions.  It  follows,  therefore,  that  we  are  not  justified  in  apply- 
ing these  formulas  to  concentrated  solutions.  Up  to  the  present  time 
we  have  no  satisfactory  theory  of  concentrated  solutions,  neither 
can  we  state  up  to  what  concentration  the  gas  laws  apply. 

PROBLEMS. 

1.  At  10°  C.  the  osmotic  pressure  of  a  solution  of  urea  is  500  mm. 
of  mercury.     If  the  solution  is  diluted  to  ten  times  its  original  volume, 
what  is  the  osmotic  pressure  at  15°  C.?  Ans.   50.89  mm. 

2.  The  osmotic  pressure  of  a  solution  of  0.184  gram  of  urea  in  100  cc. 
of  water  was  56  cm.  of  mercury  at  30°  C.     Calculate  the  molecular  weight 
of  urea.  Ans.   62.12. 

3.  At  24°  C.  the  osmotic  pressure  of  a  cane  sugar  solution  is  2.51  atmos- 
pheres.    What  is  the  concentration  of  the  solution  in  mols  per  liter? 

Ans.  0.103. 

4.  At  25°.  1  C.  the  osmotic  pressure  of  solution  of  glucose  containing 
18  grams  per  liter  was  2.43  atmospheres.     Calculate  the  numerical  value 
of  the  constant  R,  when  the  unit  of  energy  is  the  gram-centimeter. 

Ans.  84,231. 

5.  The  vapor  pressure  of  ether  at  20°  C.  is  442  mm.  and  that  of  a  solu- 
tion of  6.1  grams  of  benzoic  acid  in  50  grams  of  ether  is  410  mm.  at  the 
same  temperature.     Calculate  the  molecular  weight  of  benzoic  acid  in 
ether.  Ans.   124. 


224  THEORETICAL  CHEMISTRY 

6.  At  10°  C.  the  vapor  pressure  of  ether  is  291.8  mm.  and  that  of  a 
solution  containing  5.3  grams  of  benzaldehyde  in  50  grams  of  ether  is 
271.8  mm.    What  is  the  molecular  weight  of  benzaldehyde? 

Ans.   106.6. 

7.  A  solution  containing  0.5042  gram  of  a  substance  dissolved  in  42.02 
grams  of  benzene  boils  at  80°.  175  C.     Find  the  molecular  weight  of  the 
solute,  having  given  that  the  boiling-point  of  benzene  is  80°.00  C.,  and  its 
heat  of  vaporization  is  94  calories  per  gram.  Ans.   181.9. 

8.  A  solution  containing  0.7269  gram  of  camphor  (mol.  wt.  =  152) 
in  32.08  grams  of  acetone  (boiling-point  =  56°.30  C.)  boiled  at  56°.55  C. 
What  is  the  molecular  elevation  of  the  boiling-point  of  acetone?    What  is 
its  heat  of  vaporization? 

Ans.  K  =  16.74;  w  =  129.5  cals.  per  gm. 

9.  A  solution  of  9.472  grams  of  CdI2  in  44.69  grams  of  water  boiled  at 
100°.303  C.    The  heat  of  vaporization  of  water  is  536  calories  per  gram. 
What  is  the  molecular  weight  of  CdI2  in  the  solution?    What  conclusion 
as  to  the  state  of  CdI2  in  solution  may  be  drawn  from  the  result? 

Ans.  363.2. 

10.  The  freezing-point  of  pure  benzene  is  5°.440  C.  and  that  of  a  solu- 
tion containing  2.093  grams  of  benzaldehyde  in  100  grams  of  benzene  is 
4°.440  C.    Calculate  the  molecular  weight  of  benzaldehyde  in  the  solu- 
tion.   K  for  benzene  is  50.  Ans.    104.6. 

11.  A  solution  of  0.502  gram  of  acetone  in  100  grams  of  glacial  acetic 
acid  gave  a  depression  of  the  freezing-point  of  0°.339  C.     Calculate  the 
molecular  depression  for  glacial  acetic  acid.  Ans.   39. 

12.  By  dissolving  0.0821  gram  of  m-hydroxybenzaldehyde  (C7H602)  in 
20  grams  of  naphthalene  (melting  point  80°.  1  C.)  the  freezing-point  is 
lowered  by  0°.232  C.    Assuming  that  the  molecular  weight  of  the  solute 
is  normal  in  the  solution,  calculate  the  molecular  depression  for  naphtha- 
lene and  the  heat  of  fusion  per  gram. 

Ans.  K  =  68.96;    w  =  36.2  cals.  per  gm. 


CHAPTER  XI. 
ASSOCIATION,  DISSOCIATION  AND   SOLVATION. 

Abnormal  Solutes.  As  has  already  been  pointed  out  the 
acceptance  of  Avogadro's  hypothesis  was  greatly  retarded  by  the 
discovery  of  certain  substances  whose  vapor  densities  were  ab- 
normal. Thus,  the  vapor  density  of  ammonium  chloride  is  approx- 
imately one-half  of  that  required  by  the  formula  NH4C1,  while 
the  vapor  density  of  acetic  acid  corresponds  to  a  formula  whose 
molecular  weight  is  greater  than  that  calculated  from  the  formula, 
C2H4O2.  The  anomalous  behavior  of  ammonium  chloride  and 
kindred  substances  has  been  shown  to  be  due,  not  to  a  failure  of 
Avogadro's  law,  but  to  a  breaking  down  of  the  molecules,  —  a 
process  known  as  dissociation.  The  abnormally  large  molecular 
weight  of  acetic  acid  on  the  other  hand,  has  been  ascribed  to  a 
process  of  aggregation  of  the  normal  molecules,  known  as  asso- 
ciation. In  extending  the  gas  laws  to  dilute  solutions  similar 
phenomena  have  been  encountered. 

Association  in  Solution.  When  the  molecular  weight  of  acetic 
acid  in  benzene  is  determined  by  the  freezing-point  method,  the 
depression  of  the  freezing-point  is  abnormally  small  and  conse- 
quently, as  the  formula 


shows,  the  molecular  weight  will  be  greater  than  that  correspond- 
ing to  the  formula,  C2H4O2.  Acetic  acid  in  benzene  solution  is 
thus  shown  to  be  associated.  Almost  all  compounds  containing 
the  hydroxyl  and  cyanogen  groups  when  dissolved  in  benzene  are 
found  to  be  associated.  Solvents,  such  as  benzene  and  chloro- 
form, are  frequently  termed  associating  solvents,  although  it  is 
doubtful  whether  they  exert  any  associating  action.  There  is 
considerable  experimental  evidence  to  show  that  those  substances 

225 


226  THEORETICAL  CHEMISTRY 

whose  molecules  are  associated  in  benzene  and  chloroform  solu- 
tion, are  also  associated  in  the  free  condition.  Just  as  the  depres- 
sion of  the  freezing-point  of  a  solution  of  an  associated  substance 
is  abnormally  small,  so  its  osmotic  pressure  and  other  related 
properties  will  be  less  than  the  calculated  values. 

Dissociation  in  Solution.  Van't  Hoff  *  pointed  out  that  the 
osmotic  pressure  of  solutions  of  most  salts,  of  all  strong  acids,  and 
of  all  strong  bases  is  much  greater  for  all  concentrations  than 
would  be  expected  from  the  osmotic  pressure  of  solutions  of  sub- 
stances, like  cane  sugar  or  urea,  for  corresponding  concentrations. 
He  was  unable  to  account  for  this  abnormal  behavior,  and  in  order 
to  render  the  general  gas  equation  applicable,  he  introduced  a 
factor  i,  the  modified  equation  being 

PV  =  iRT. 

If  the  osmotic  pressure  of  some  substance,  like  cane  sugar,  which 
behaves  normally,  be  represented  by  P0>  the  factor  i  is  given  by 
the  expression 

.      P 


Since  the  osmotic  pressure  of  a  solution  is  proportional  to  the 
relative  lowering  of  its  vapor  pressure,  to  the  elevation  of  its 
boiling-point,  and  to  the  lowering  of  its  freezing-point,  we  may 
write 


Po 


where  the  symbols  have  their  usual  significance.  The  subscript 
0  refers  in  each  case  to  a  substance  which  behaves  normally,  and  A 
denotes  either  boiling-point  elevation  or  freezing-point  depression. 
A  more  definite  conception  of  the  abnormal  behavior  of  salts 
will  be  gained  by  an  inspection  of  the  accompanying  tables.  In 
the  first  column  is  recorded  the  molar  concentration  of  the  solu- 
tion; the  second  column  gives  the  observed  depressions  of  the 

*  Zeit.  phys.  Chem.,  i,  501  (1887). 


ASSOCIATION,   DISSOCIATION  AND  SOLVATION          227 

freezing-point  and  the  third  column  contains  the  values  of  the 
ratio  of  the  observed  depression  to  the  normal  depression,  or  i. 


POTASSIUM  CHLORIDE. 


POTASSIUM  SULPHATE. 


TO 

A 

i 

m 

A 

i 

0.05 

0.1750 

1.88 

0.05 

0.2270 

2.33 

0.10 

0.3445 

1.85 

0.10 

0.4317 

2.32 

0.20 

0.6808 

1.83 

0.20 

0.8134 

2.18 

0.40 

1.3412 

1.80 

0.30 

1  .  1673 

2.09 

ALUMINIUM  CHLORIDE. 


SODIUM  CHROMATE. 


TO 

A 

i 

TO 

A 

i 

0.046 
0.076 
0.102 

0.276 
0.446 
0.578 

3.22 
3.15 
3.04 

0.1 
0.2 
0.5 
1.0 

0.450 
0.850 
1.960 
3.800 

2.42 
2.28 
2.11 
2.04 

It  is  apparent  from  the  above  data  that  the  depression  of  the 
freezing-point  of  water  caused  by  these  salts  is  abnormally  large, 
a  fact  which  points  to  an  increase  in  the  number  of  dissolved  units 
over  that  corresponding  to  the  initial  concentration. 

The  Theory  of  Electrolytic  Dissociation.  In  1887,  Arrhenius  * 
advanced  an  hypothesis  to  account  for  the  abnormal  osmotic 
activity  of  solutions  of  acids,  bases  and  salts.  He  pointed  out 
that  just  as  the  exceptional  behavior  of  certain  gases  has  been 
completely  reconciled  with  the  law  of  Avogadro,  by  assuming  a 
dissociation  of  the  vaporized  molecule  into  two  or  more  simpler 
molecules,  so  the  enhanced  osmotic  pressure  and  the  abnormally 
great  freezing-point  depression  of  solutions  of  acids,  bases  and 
salts  can  be  explained,  if  we  assume  a  similar  process  of  dissoci- 
ation. He  proposed,  therefore,  that  aqueous  solutions  of  acids, 
bases  and  salts  be  considered  as  dissociated,  to  a  greater  or  less 
extent,  into  positively-  and  negatively-charged  particles  or  ions, 
and  that  the  increase  in  the  number  of  dissolved  units  due  to  this 

*  Zeit.  phys.  Chem.,  i,  631  (1887). 


228  THEORETICAL  CHEMISTRY 

dissociation  is  the  cause  of  the  enhanced  osmotic  activity.  Accord- 
ing to  this  hypothesis,  hydrochloric  acid,  potassium  hydroxide  and 
potassium  chloride,  when  dissolved  in  water,  dissociate  in  the 
following  manner:  — 

TT/^l  _v  TI*       I       OT 

XlV^l      — r  XI     -J-  \jL 


KC1   ->K'  +  C1', 

where  the  dots  indicate  positively-charged  ions  and  the  dashes 
negatively-charged  ions. 

In  each  of  the  above  cases,  one  molecule  yields  two  ions,  so 
that,  if  dissociation  is  complete,  the  maximum  osmotic  effect 
should  not  be  greater  than  twice  that  produced  by  an  equimolecu- 
lar  quantity  of  a  substance  which  behaves  normally.  Reference 
to  the  preceding  table  shows  that  the  value  of  i  for  potassium 
chloride  approaches  the  limiting  value  of  2  as  the  solution  is 
diluted.  The  other  salts  given  in  the  table  dissociate,  according 
to  Arrhenius,  in  the  following  way :  — - 

K2S04-^2K--hS04" 
NasCr04  ->  2  Na*  +  Cr04" 
Aid.  ->  A!"'  +  3  Cl'. 

If  these  equations  correctly  represent  the  process  of  dissociation, 
then  when  dissociation  is  complete,  the  osmotic  effect  of  infinitely 
dilute  solutions  of  potassium  sulphate  and  sodium  chromate 
should  be  three  times  the  effect  produced  by  an  equimolecular 
quantity  of  a  normal  solute,  and  in  the  case  of  aluminium  chloride, 
the  maximum  effect  should  be  four  times  the  effect  due  to  a  normal 
substance.  A  glance  at  the  table  shows  that  the  values  of  i  for 
solutions  of  the  three  salts  approach  these  respective  limits.  If 
this  hypothesis  of  ionic  dissociation  be  accepted,  then  it  becomes 
possible  to  calculate  the  degree  of  ionization  in  any  solution  by 
comparing  its  freezing-point  depression  with  the  freezing-point 
lowering  of  an  equimolecular  solution  of  a  normal  substance. 

Let  us  suppose  that  the  degree  of  dissociation  of  1  molecule  of 
a  dissolved  substance  is  a,  each  molecule  yielding  n  ions.  Then 


ASSOCIATION,   DISSOCIATION  AND  SOLVATION         229 


1  —  a.  will  be  the  undissociated  portion  of  the  molecule,  and  the 
total  number  of  dissolved  units  will  be 

1  —  a.  -f-  na. 

If  A  is  the  depression  of  the  freezing-point  produced  by  the  sub- 
stance,  and   Ao  the  depression  produced  by  an  equimolecular 
quantity  of  an  undissociated  substance,  then 
1  —  a  +  na       A 


or 


1 


a 


i-  1 


n  -  1 

It  will  be  observed  that  this  formula  is  identical  with  that  derived 
for  the  degree  of  dissociation  of  a  gas  (p.  91).  If  this  formula 
be  applied  to  the  freezing-point  data  for  solutions  of  potassium 
chloride  given  in  the  preceding  table  we  find  the  following  per- 
centages of  dissociation  corresponding  to  the  different  concentra- 
tions :  — 

POTASSIUM  CHLORIDE. 


m 

A 

i 

a 

0.05 

0.1750 

1.88 

88 

0.10 

0.3445 

1.85 

85 

0.20 

0.6808 

1.83 

83 

0.40 

1.3412 

1.80 

80 

The  figures  in  the  last  column  show  that  the  degree  of  dissocia- 
tion increases  as  the  concentration  diminishes.  It  was  further 
pointed  out  by  Arrhenius  that  all  of  the  substances  which  exhibit 
abnormal  osmotic  effects,  when  dissolved  in  water,  yield  solutions 
which  conduct  the  electric  current,  whereas,  aqueous  solutions  of 
such  substances  as  cane  sugar,  urea  and  alcohol,  exert  normal 
osmotic  pressures,  but  do  not  conduct  electricity  any  better  than 
the  pure  solvent.  In  other  words,  only  electrolytes  *  are  capable  of 
undergoing  ionic  dissociation ;  hence  Arrhenius  termed  his  hypoth- 
esis the  electrolytic  dissociation  theory.  As  we  have  seen,  when 
potassium  chloride  is  dissolved  in  water,  it  is  supposed  to  dissoci- 

*  The  term  electrolyte  strictly  refers  to  the  solution  of  an  ionized  substance, 
although  it  is  often  applied  to  acids,  bases  and  salts  because,  when  dissolved, 
they  produce  electrolytes.  To  avoid  confusion,  the  term  "ionogen"  (ion 
former)  has  been  proposed  for  those  substances  which  give  conducting  solu- 
tions. 


230 


THEORETICAL  CHEMISTRY 


ate  into  positively-charged  potassium  ions  and  negatively-charged 
chlorine  ions.  Accordingly  when  two  platinum  electrodes,  one 
charged  positively  and  the  other  negatively,  are  introduced  into 
the  solution,  the  potassium  ions  move  toward  the  negative  elec- 
trode and  the  chlorine  ions  move  toward  the  positive  electrode,  the 
passage  of  a  current  through  the  solution  consisting  in  the  ionic 
transfer  of  electric  charges.  Since  the  undissociated  molecules, 
being  electrically  neutral,  do  not  participate  in  the  transfer  of 
electric  charges,  it  follows  that  the  conductance  of  a  solution  of 
an  electrolyte  is  dependent  upon  the  degree  of  dissociation.  The 
relation  between  electrical  conductance  and  the  degree  of  ioniza- 
tion  will  be  discussed  in  a  subsequent  chapter.  It  may  be  stated 
at  this  point,  however,  that  the  values  of  a  based  upon  measure- 
ments of  electrical  conductance,  while  showing  some  discrepancies 
in  individual  cases,  are  in  general  in  good  agreement  with  the 
values  obtained  by  the  freezing-point  method.  Furthermore,  the 
values  of  a  obtained  from  freezing-point  measurements  are  in  har- 
mony with  those  based  upon  De  Vries'  isotonic  coefficients. 
It  will  be  seen,  on  referring  to  the  table  of  isotonic  coefficients 
(p.  181),  that  solutions  of  electrolytes  show  enhanced  osmotic 
activity.  Thus,  the  osmotic  pressures  of  equi-molecular  solutions 
of  cane  sugar,  potassium  nitrate  and  calcium  chloride  are  to  each 
other  as  1  :  1.67  :  2.40. 

The  following  table  illustrates  the  general  agreement  between 
the  values  of  i  calculated,  (a)  from  electrical  conductance,  (b) 
from  freezing-point  depression,  and  (c)  from  De  Vries'  isotonic 
coefficients. 


Substance. 

Molar  Cone. 

(a) 

(b) 

(c) 

KC1.. 

0   14 

1  86 

1  82 

1  81 

LiCl  

0  13 

1  84 

1  94 

1  92 

Ca(NO3)2 

0  18 

2  46 

2  47 

2  48 

MgCl2 

0  19 

2  48 

2  68 

2  79 

CaCl2  

0  184 

2  42 

2  67 

2  78 

It  must  be  remembered  that  the  values  of  i  derived  from  freez- 
ing-point   measurements    correspond    to    temperatures    in    the 


ASSOCIATION,   DISSOCIATION  AND  SOLVATION          231 

neighborhood  of  0°  C.,  while  those  derived  from  the  other 
methods  correspond  to  temperatures  ranging  from  17°  C.  to 
25°  C. 

Chemical  Properties  of  Completely  Ionized  Solutions.  The 
chemical  properties  of  an  ion  are  very  different  from  the  properties 
of  the  atom  or  radical  when  deprived  of  its  electrical  charge. 
For  example,  the  sodium  ion  is  present  in  an  aqueous  solution  of 
sodium  chloride,  but  there  is  no  evidence  of  chemical  reaction  with 
the  solvent;  whereas,  the  element  in  the  electrically-neutral  con- 
dition reacts  violently  with  water,  evolving  hydrogen  and  form- 
ing a  solution  of  potassium  hydroxide.  Again,  take  the  element 
chlorine:  when  chlorine  in  the  molecular  condition,  either  as  gas 
or  in  solution,  is  added  to  a  solution  of  silver  nitrate,  no  precipitate 
of  silver  chloride  is  formed.  Further,  chlorine  in  such  compounds 
as  CHC13,  CC14,  etc.,  is  not  precipitated  by  silver  nitrate,  since 
these  compounds  are  not  dissociated  by  water.  Or,  chlorine  may 
be  present  in  a  compound  which  is  dissociated  by  water  and  yet 
not  exhibit  its  characteristic  reactions,  because  it  is  present  in  a 
complex  ion.  Thus,  potassium  chlorate  dissociates  in  the  follow- 
ing manner  :  — 

KClOa  -»  K'  + 


On  adding  silver  nitrate,  there  is  no  precipitation,  because  the 
chlorine  forms  a  complex  ion  with  oxygen. 

Physical  Properties  of  Completely  Ionized  Solutions.  The 
physical  properties  of  completely  ionized  solutions  are,  in  general, 
additive.  This  is  well  illustrated  by  a  series  of  solutions  of 
colored  salts,  the  color  of  which  is  due  to  the  presence  of  a  par- 
ticular ion.  It  is  found,  when  the  solutions  are  sufficiently 
dilute  to  insure  complete  dissociation,  that  they  all  have  the 
same  color.  The  additive  character  of  the  colors  of  solutions  of 
electrolytes  is  brought  out  in  a  striking  manner  by  a  comparison 
of  their  absorption  spectra.  Ostwald  *  photographed  the  absorp- 
tion spectra  of  solutions  of  the  permanganates  of  lithium,  cadmium, 
ammonium,  zinc,  potassium,  nickel,  magnesium,  copper,  hydrogen 
and  aluminium,  each  solution  containing  0.002  gram-equivalents 

*  Zeit.  phys.  Chem.,  9,  579  (1892). 


232 


THEORETICAL  CHEMISTRY 


of  salt  per  liter.  The  absorption  spectra,  as  shown  in  Fig.  67, 
will  be  seen  to  be  practically  identical,  the  bands  occupying  the 
same  position  in  each  spectrum.  This  affords  a  strong  con- 
firmation of  the  theory  of  electrolytic  dissociation,  according  to 

which  a  dilute  solution  is  to  be 
regarded  as  a  mixture  of  elec- 
trically equivalent  quantities  of 
oppositely  charged  ions,  each  of 
which  contributes  its  specific 
properties  to  the  solution.  The 
permanganate  ion  being  colored, 
and  common  to  all  of  the  salts 
examined,  and  the  positive  ions 
of  the  various  substances  being 
colorless,  it  follows  that  when  dis- 
sociation is  complete,  the  absorp- 
tion spectra  of  all  of  the  solutions 
must  be  identical.  A  number  of 
other  properties  of  completely 
dissociated  solutions  have  been 
shown  to  be  additive.  Among 
these  may  be  mentioned  density, 
specific  refraction,  surface  ten- 
sion, thermal  expansion,  and 
magnetic  rotatory  power.  Addi- 
tional evidence  in  favor  of  the 
theory  of  electrolytic  dissociation 
will  be  furnished  in  forthcoming 
chapters.  Notwithstanding  the 
large  number  of  facts  which  can 
be  satisfactorily  interpreted  by  the  theory,  there  are  directions 
in  which  it  requires  amplification  and  modification.  Of  the 
various  objections  which  have  been  urged  against  the  theory  of 
electrolytic  dissociation,  one  is  of  sufficient  weight  to  call  for 
brief  consideration  here.  When  two  elements,  such  as  potassium 
and  chlorine,  combine  to  form  potassium  chloride,  the  reaction 
is  violent  and  a  large  amount  of  heat  is  developed.  Nevertheless, 


Fig.  67. 


ASSOCIATION,   DISSOCIATION  AND  SOLVATION          233 

according  to  the  ionization  theory,  the  strong,  mutual  affinity  of 
these  two  elements  is  overcome  by  the  act  of  solution  in  water, 
the  molecule  being  split  into  two  oppositely-charged  atoms.  Obvi- 
ously such  a  separation  calls  for  the  expenditure  of  a  large  amount 
of  energy,  and  the  question  naturally  arises :  —  What  is  the 
source  of  this  energy?  While  this  question  cannot  be  fully  an- 
swered here,  it  may  be  pointed  out  that  we  have  abundant 
evidence  to  show  that  the  ions  are  hydrated,  each  being  surrounded 
by  an  " atmosphere"  of  solvent.  In  view  of  this  fact,  it  has  been 
suggested  *  that  dissociation  in  aqueous  solution  is  caused  by  the 
mutual  attraction  between  the  ions  and  the  molecules  of  the  sol- 
vent, the  heat  of  ionic  hydration  furnishing  the  energy  necessary 
for  the  separation  of  the  ions. 

Freezing-Point  Depressions  Produced  by  Concentrated  Solu- 
tions of  Electrolytes.  As  has  already  been  mentioned,  the 
dissociation  of  electrolytes  in  aqueous  solution  increases  with  the 
dilution,  becoming  complete  at  a  concentration  of  about  0.001 
molar.  We  should  expect  the  dissociation  to  diminish  with  in- 
creasing concentration,  until,  if  the  electrolyte  is  sufficiently  solu- 
ble, the  depression  of  the  freezing-point  becomes  normal.  Recent 
investigations  by  Jones  and  his  co-workers  f  have  shown  that  the 
facts  are  contradictory  to  this  expectation.  They  found  that  the 
value  of  the  molecular  depression  of  the  freezing-point  of  water 
produced  by  a  number  of  chlorides  and  bromides,  diminished  with 
increasing  concentration  up  to  a  certain  point,  as  would  be  ex- 
pected, and  then  increased  again.  The  increase  in  the  molecular 
depression  became  very  marked  at  great  concentrations;  in  fact, 
the  molecular  depression  in  a  molar  solution  was  frequently  greater 
than  the  molecular  depression  corresponding  to  a  completely 
dissociated  salt.  This  phenomenon  was  systematically  studied 
by  Jones  and  the  author  J  and  the  fact  was  established  that  it 
is  quite  general. 

*  Trans.  Faraday  Soc.,  i,  197  (1905);  3,  123  (1907). 

f  Am.  Chem.  Jour.,  22,  5,  110  (1899);  23,  89  (1900). 

j  Zeit.  phys.  Chem.,  46,  244  (1903);  Phys.  Rev.,  18,  146  (1904):  Am.  Chem. 
Jour.,  31,  303  (1904);  32,  308  (1904);  33,  534  (1905);  34,  291  (1905);  Zeit. 
phys.  Chem.,  49,  385  (1904);  Monograph  No.  60,  Carnegie  Institution  of 
Washington. 


234  THEORETICAL  CHEMISTRY 

This  abnormal  depression  of  the  freezing-point  may  be  accounted 
for  by  assuming  that  the  dissolved  substance  has  entered  into  com- 
bination with  a  portion  of  the  water,  thus  removing  it  from  the 
role  of  solvent.  The  formation  of  a  loose  molecular  complex  be- 
tween one  molecule  of  the  solute  and  a  large  number  of  molecules 
of  water,  acts  as  a  single  dissolved  unit  in  depressing  the  freezing- 
point  of  the  pure  solvent.  Evidently  the  total  amount  of  water 
present,  which  functions  as  solvent,  is  diminished  by  the  amount 
of  water  which  has  been  appropriated  by  the  solute.  The  abnor- 
malities observed  in  the  depression  of  the  freezing-point  of  con- 
centrated solutions  of  electrolytes  can  be  explained  by  assuming 
that  the  molecules  of  solute,  or  the  resulting  ions,  are  in  combina- 
tion with  a  number  of  molecules  of  solvent.  This  hypothesis  is 
termed  the  solvate  theory,  and  the  loose  molecular  complexes  are 
called  solvates.  Since  the  solvate  theory  was  first  proposed,  con- 
siderable evidence  has  been  accumulated  to  confirm  its  correct- 
ness. Reference  has  already  been  made  to  the  work  of  Philip  on 
the  solubility  of  gases  in  saline  solutions,  from  which  he  concludes 
that  the  dissolved  salts  enter  into  combination  with  a  portion  of 
the  solvent.  The  experiments  of  Morse  and  the  Earl  of  Berkeley 
on  osmotic  pressure,  also  seem  to  point  to  the  solvation  of  the 
dissolved  substance. 

PROBLEMS. 

1.  At  18°  C.  a  0.5  molar  solution  of  XaCl  is  74.3  per  cent  dissociated. 
What  would  be  the  osmotic  pressure  of  the  solution  in  atmospheres  at 
18°  C.?  Ans.  20.79. 

2.  A  solution  containing  3  mols  of  cane  sugar  per  liter  was  found  by 
the  plasmolytic  method  to  be  isotonic  with  a  solution  of  potassium  nitrate 
containing  1.8  mols  per  liter.     What  is  the  degree  of  ionization  of  the 
potassium  nitrate?  Ans.  67  per  cent. 

3.  The  vapor  pressure  of  water  at  20°  C.  is  17.406  mm.  and  that  of  a 
0.2  molar  solution  of  potassium  chloride  u  17.296  mm.  at  the  same  tem- 
perature.   Calculate  the  degree  of  dissociation  of  the  salt. 

Ans..  75.38  per  cent. 

4.  The  degree  of  dissociation  of  a  0.5  molar  solution  of  sodium  chloride 
at  25°  is  74.3  per  cent.     Calculate  the  osmotic  pressure  of  the  solution  at 
the  same  temperature.  Ans.  21.32  atmos. 


ASSOCIATION,   DISSOCIATION  AND  SOLVATION          235 

5.  A  solution  containing  1.9  mols  of  calcium  chloride  per  liter  is  isotonic 
with  a  solution  of  glucose  containing  4.05  mols  per  liter.    What  is  the 
degree  of  ionization  of  the  calcium  chloride?  Ans.   56.6  per  cent. 

6.  At  0°  C.  the  vapor  pressure  of  water  is  4.620  mm.  and  of  a  solution 
of  8.49  grams  of  NaNO3  in  100  grams  of  water  4.483  mm.     Calculate  the 
degree  of  ionization  of  NaN03.  Ans.   64.9  per  cent. 

7.  At  0°  C.  the  vapor  pressure  of  water  is  4.620  mm.  and  that  of  a 
solution  of  2.21  grams  of  CaCl2  in  100  grams  of  water  is  4.583  mm.     Cal- 
culate the  apparent  molecular  weight  and  the  degree  of  ionization  of 
CaCl2.  Ans.   M  =  49.66,     a  =  62  per  cent. 

8.  The  boiling-point  of  a  solution  of  0.4388  gram  of  sodium  chloride 
in  100  grams  of  water  is  100°. 074  C.     Calculate  the  apparent  molecular 
weight  of  the  sodium  chloride  and  its  degree  of  ionization.    K  =  5.2. 

Ans.   M  =  30.84,     a  =  89.7  per  cent. 

9.  The  boiling-point  of  a  solution  of  3.40  grams  of  BaCk  in  100  grams 
of  water  is  100°.208  C.    K  =  5.2.    What  is  the  degree  of  ionization  of 
theBaCl2?  Ans.   72.5  per  cent. 

10.  At  100°  C.  the  vapor  pressure  of  a  solution  of  6.48  grams  of  ammo- 
nium chloride  in  100  grams  of  water  is  731.4  mm.     K  =  5.2.     What  is  the 
boiling-point  of  the  solution?  Ans.    101°.086  C. 

11.  A  solution  of  1  gram  of  silver  nitrate  in  50  grams  of  water  freezes 
at  —  0°.348  C.     Calculate  to  what  extent  the  salt  is  ionized  in  solution. 
K  =  18.6.  Ans.   59  per  cent. 

12.  A  solution  of  NaCl  containing  3.668  grams  per  1000  grams  of 
water  freezes  at  —  0°.2207  C.     Calculate  the  degree  of  ionization  of  the 
salt.    K  =  18.6.  Ans.   89.2  per  cent. 

13.  The  freezing-point  of  a  solution  of   barium  hydroxide  containing 
1  mol  in  64  liters  is  —  0°.0833  C.     What  is  the  concentration  of  hydroxyl 
ions  in  the  solution?    Take  K  =  18.9  for  concentrations  in  mols  per 
liter.  Ans.   0.0284  gm.-ion  per  liter. 

14.  The  vapor  pressure  of  water  at  0°  C.  is  4.620  mm.,  and  the  lower- 
ing of  the  vapor  pressure  produced  by  dissolving  5.64  grams  of  sodium 
chloride  in  100  grams  of  water  is  0.142  mm.     What  is  the  freezing-point 
of  the  solution?    K  =  18.6.  Ans.    -  3°.177  C. 

15.  A  solution  containing  8.34  grams  Na2S04  per  1000  grams  of  water 
freezes  at  —  0°.280  C.     Assuming  dissociation  into  3  ions,  calculate  the 


236  THEORETICAL  CHEMISTRY 

degree  of  ionization  and  the  concentrations  of  the  Na*  and  SO/'  ions. 
K  =  18.6. 

Ans.   a  =  78.2  per  cent;   cone.  Na"  =  0.0918  gm.-ion  per  liter;  cone. 
S04"  =  0.0459  gm.-ion  per  liter. 


CHAPTER  XII. 
COLLOIDS. 

Crystalloids  and  Colloids.  In  the  course  of  his  investigations 
on  diffusion  in  solutions,  Thomas  Graham  *  drew  a  distinction 
between  two  classes  of  solutes,  which  he  termed  crystalloids  and 
colloids.  Crystalloids,  as  the  name  implies,  can  be  obtained  in  the 
crystalline  form:  to  this  class  belong  nearly  all  of  the  acids,  bases 
and  salts.  Colloids,  on  the  other  hand,  are  generally  amorphous, 
such  substances  as  albumin,  starch  and  caramel  being  typical  of 
the  class.  Because  of  the  gelatinous  character  of  many  of  the  sub- 
stances in  this  class,  Graham  termed  them  colloids  (xoXXa  =  glue, 
and  erdos=  form).  The  differences  between  the  two  classes  are 
most  apparent  in  the  physical  properties  of  their  solutions.  Thus, 
crystalloids  diffuse  much  more  rapidly  than  colloids;  the  velocity 
of  diffusion  of  caramel  being  nearly  100  times  slower  than  that  of 
hydrochloric  acid  at  the  same  temperature.  While  crystalloids 
exert  osmotic  pressure,  lower  the  vapor  pressure  and  depress  the 
freezing-point  of  the  solvent,  colloids  have  very  little  effect  upon 
the  properties  of  the  solvent.  The  marked  differences  in  the  rates 
of  diffusion  of  crystalloids  and  colloids  render  their  separation 
comparatively  easy.  If  a  solution  containing  both  crystalloids 
and  colloids  be  placed  in  a  vessel  over  the  bottom  of  which  is 
stretched  a  colloidal  membrane,  such  as  parchment,  and  the  whole 
is  immersed  in  pure  water,  the  crystalloids  will  pass  through  the 
membrane,  while  the  colloids  will  be  left  behind.  This  process 
was  termed  by  Graham,  dialysis,  while  the  apparatus  employed  to 
effect  such  a  separation  was  called  a  dialyzer.  When  a  solution  of 
sodium  silicate  is  added  to  an  excess  of  hydrochloric  acid,  the 
products  of  the  reaction,  silicic  acid  and  sodium  chloride,  remain 
in  solution.  When  the  mixture  is  placed  in  a  dialyzer,  the  sodium 
chloride  and  the  hydrochloric  acid,  being  crystalloids,  diffuse 

*  Lieb.  Ann.,  121,  1  (1862). 
237 


238  THEORETICAL  CHEMISTRY 

through  the  membrane  of  the  dialyzer,  leaving  behind  the  colloidal 
silicic  acid. 

The  terms  crystalloid  and  colloid,  as  used  at  the  present  time, 
have  acquired  different  meanings  from  those  assigned  to  them  by 
Graham.  The  terms  are  now  considered  to  refer,  not  so  much  to 
different  classes  of  substances,  as  to  different  states  which  almost 
all  substances  can  assume  under  certain  conditions. 

Colloidal  Solutions.  A  colloidal  solution  is  one  in  which  the 
solute  is  a  colloid,  although  the  latter  may  not  be  included  among 
the  substances  classified  as  such  by  Graham.  For  example, 
arsenious  sulphide,  ferric  hydroxide  or  finely-divided  gold  may 
form  colloidal  solutions.  In  bringing  such  substances  into  the 
colloidal  state,  mere  agitation  with  water  will  not  suffice,  but  some 
indirect  method  must  be  employed. 

Nomenclature.  Graham  distinguished  between  two  condi- 
tions in  which  colloids  were  obtainable,  the  term  sol  being  applied 
to  forms  in  which  the  system  resembled  a  liquid,  while  the  term 
gel  was  used  to  designate  those  forms  which  were  solid  and  jelly- 
like.  When  one  of  the  components  of  the  solution  was  water,  the 
two  forms  were  called  a  hydrosol  and  a  hydrogel.  In  like  manner, 
when  alcohol  was  one  of  the  components,  the  terms  alcosol  and 
alcogel  were  applied  to  the  two  forms. 

As  the  knowledge  of  colloids  has  developed  it  has  become  neces- 
sary to  supplement  Graham's  nomenclature  by  the  introduction 
of  various  other  terms.  It  is  known  to-day  that  the  essential 
difference  between  colloidal  suspensions  and  solutions  on  the  one 
hand,  and  true  solutions  on  the  other,  is  due  to  the  difference  in  the 
degree  of  subdivision  or  degree  of  dispersity  of  the  dissolved  sub- 
stance. In  a  true  solution  the  dissolved  substance  is  generally 
present  either  in  the  molecular  or  ionic  condition,  as  may  be  shown 
by  means  of  the  familiar  osmotic  methods  for  molecular  weight  de- 
termination. In  colloidal  solutions,  however,  the  degree  of  dis- 
persity is  not  so  great  and  has  been  found  to  vary  from  above  the 
limit  of  microscopic  visibility  (1  X  10~5  cm.)  to  that  of  molecular 
dimensions  (1  X  10~8  cm.).  When  the  degree  of  dispersity  varies 
from  1  X  10~3  cm.  to  1  X  10~5  cm.  the  particles  are  termed 
microns.  The  properties  of  the  disperse  phase  at  this  degree  of 


COLLOIDS  239 

dispersity  differ  appreciably  from  the  properties  of  the  same  sub- 
stance when  present  in  large  masses.  When  the  degree  of  dis- 
persity lies  between  1  X  10~5  cm.  and  5  X  10~7  cm.  the  particles 
are  known  as  submicrons.  The  existence  of  particles  whose  diam- 
eters are  approximately  1  'X  10~7  cm.  has  been  demonstrated  by 
Zsigmondy  with  the  ultramicroscope;  these  minute  particles  are 
termed  amicrons.  When  the  degree  of  dispersity  is  increased 
beyond  this  limit  all  heterogeneity  apparently  vanishes  and  we 
enter  the  realm  of  true  solutions. 

When  the  dispersion  is  not  too  great,  colloidal  solutions  may  be 
divided  into  suspensions  and  emulsions  according  to  whether  the 
disperse  phase  is  a  solid  or  a  liquid.  As  the  dispersion  is  increased 
we  obtain  suspension  and  emulsion  colloids  which  may  be  con- 
veniently called  suspensoids  and  emulsoids.  Suspensoids  and 
emulsoids  are  included  under  the  general  term  dispersoids.  In 
certain  cases,  although  the  disperse  phase  is  unquestionably  liquid, 
the  systems  resemble  suspensoids  in  their  behavior,  while  in  other 
cases,  where  the  disperse  phase  is  solid,  the  systems  exhibit  proper- 
ties characteristic  of  emulsoids.  For  this  reason  the  classification 
of  sols  as  suspensoids  and  emulsoids  is  not  entirely  satisfactory. 
A  better  system  is  that  in  which  the  presence  or  the  absence  of 
affinity  between  the  disperse  phase  and  the  dispersion  medium  is 
made  the  basis  of  classification.  Where  there  is  marked  affinity 
between  the  two  phases,  the  system  is  termed  lyophile,  and  where 
such  affinity  is  absent,  the  system  is  termed  lyophobe.  When  the 
dispersion  medium  is  water,  the  terms  hydrophile  and  hydrophobe 
are  employed. 

In  the  reversible  transformation  of  a  sol  into  a  gel,  we  are 
not  warranted  in  referring  to  the  change  from  gel  to  sol  as  an 
act  of  solution,  for  if  the  gel  really  dissolved,  a  solution  and 
not  a  sol  would  result.  Various  terms  have  been  proposed  for 
these  reversible  transformations  but  perhaps  the  most  satisfac- 
tory are  the  terms  gelation  and  solation,  the  former  designating 
the  formation  of  a  gel  from  a  sol  and  the  latter  the  reverse 
process. 

Lyotrope  Series.  The  differences  between  lyophile  and  lyo- 
phobe sols  are  frequently  very  marked,  this  being  especially  true 


240  THEORETICAL  CHEMISTRY 

of  their  behavior  toward  chemical  reagents.  The  action  of  chemi- 
cal reagents  on  lyophobe  sols  is  almost  wholly  confined  to  the 
disperse  phase,  while  the  addition  of  reagents  to  lyophile  sols  fre- 
quently produces  a  more  marked  effect  on  the  dispersion  medium 
than  on  the  disperse  phase.  It  should  be  observed  that  the  physi- 
cal properties  of  a  lyophobe  sol  and  of  the  pure  dispersion  medium 
are  practically  identical,  while  exactly  the  reverse  is  true  of  lyo- 
phile sols.  It  is  well  known  that  the  addition  of  a  foreign  sub- 
stance to  a  reaction-mixture  frequently  exerts  a  marked  influence 
on  the  speed  of  the  reaction,  notwithstanding  the  fact  that  the 
nature  of  the  added  substance  may  be  such  as  to  render  its  partici- 
pation in  the  reaction  highly  improbable.  If  a  series  of  reagents 
are  arranged  in  the  order  of  their  influence  on  a  particular  reaction, 
it  has  been  found  that  the  same  sequence  is  preserved  when  the 
same  reagents  are  added  to  other  reactions  of  widely  different 
character.  In  some  reactions  the  reagents  may  produce  effects 
directly  opposite  to  those  which  they  produce  in  others,  but  the 
sequence  remains  unchanged.  For  example,  when  the  same  re- 
action takes  place  either  in  an  acid  or  in  an  alkaline  medium,  the 
substances  which  promote  the  reaction  when  the  medium  is  acid, 
retard  it  when  the  medium  is  alkaline,  but  the  sequence  of  the 
added  substances  remains  the  same  under  both  conditions.  These 
facts  make  it  appear  highly  probable  that  the  effects  produced  by 
the  addition  of  foreign  substances  to  a  chemical  reaction  are  to  be 
ascribed  to  the  changes  which  they  produce  in  the  pure  solvent. 
It  is  not  without  significance  that  the  same  sequence  of  reagents  is 
maintained  whether  we  observe  their  influence  on  different  chemi- 
cal reactions  or  on  certain  physical  properties  of  the  solvent,  such 
as  its  density,  viscosity,  and  surface  tension. 

The  following  examples  *  afford  a  striking  illustration  of  the 
persistence  of  the  sequence  of  reagents,  generally  known  as  the 
lyotrope  series.  The  ions  which  precede  the  formula  (H20)  reduce 
the  velocity  of  reaction  or  cause  a  diminution  in  the  magnitude  of 
the  particular  physical  property  tabulated.  The  ions  which  follow 
the  formula  (H20)  exert  the  opposite  effect. 

*  Freundlich,  Kapillarchemie,  p.  411. 


COLLOIDS  241 

1.  The  hydrolysis  of  esters  by  acids. 

Anions:  S04  (H20)  Cl  <  Br. 

Cations:  (H2O)  Li  <  Na  <  K  <  Rb  <  Cs. 

2.  The  hydrolysis  of  esters  by  bases. 

Anions:  I  >  NO3  >  Br  >  Cl  (H20)  S203  <  S04. 
Cations:  Cs  >  Rb  >  K  >  Li  (H20). 

3.  The  viscosity  of  aqueous  solutions. 

Anions:  NO3  >  Cl  (H2O)  S04  (potassium  salts). 
Cations:  Cs  >  Rb  >  K  (H2O)  Na  <  Li  (chlorides), 

4.  The  surface  tension  of  aqueous  solutions. 

(H20)  K  N03  <  Cl<  S04  <  C03. 

It  will  be  observed  that  although  the  ions  which  accelerate  the 
acid  hydrolysis  retard  the  basic  hydrolysis,  the  sequence  of  the 
ions  nevertheless  remains  the  same. 

The  Ultramicroscope.  When  a  narrow  beam  of  sunlight  is 
admitted  into  a  darkened  room,  the  dust  particles  in  its  path  are 
rendered  visible  by  the  scattering  of  the  light  at  the  surface  of  the 
particles.  If  the  air  of  the  room  is  free  from  dust,  no  shining 
particles  will  be  seen  and  the  space  is  said  to  be  " optically  void." 
When  the  particles  of  dust  are  very  minute,  the  beam  of  light 
acquires  a  bluish  tint.  The  blue  color  of  the  sky  is  thus  attrib- 
uted to  the  presence  of  extremely  fine  particles  of  dust  in  the  air 
together  with  minute  drops  of  condensed  gases  in  the  upper  regions 
of  the  atmosphere. 

The  visibility  of  a  beam  of  light  due  to  the  scattering  effect  of 
minute  particles,  is  known  as  the  Tyndall  phenomenon.  Almost 
all  colloidal  solutions  exhibit  this  phenomenon  when  a  powerful 
beam  of  light  is  passed  through  them,  thus  proving  the  presence 
of  discrete  particles  in  the  solutions. 

The  ultramicroscope  is  an  instrument  devised  by  Siedentopf 
and  Zsigmondy  *  for  the  detection  of  colloidal  particles  much  too 
small  to  be  seen  by  the  naked  eye.  A  powerful  beam  of  light 
issuing  from  a  horizontal  slit  is  brought  to  a  focus  within  the 
colloidal  solution  under  examination  by  means  of  a  microscope 
objective,  and  this  image  is  viewed  through  a  second  micro- 

*  "  Colloids  and  the  Ultramicroscope  "  by  R.  Zsigmondy.  Trans,  by 
Alexander.  John  Wiley  &  Sons,  Inc. 


242  THEORETICAL  CHEMISTRY 

scope,  the  axis  of  which  is  at  right  angles  to  the  path  of  the 
beam. 

When  examined  in  this  way  a  colloidal  solution  appears  to  be 
swarming  with  brilliantly  colored  particles,  moving  rapidly  in  a 
dark  field;  whereas  a  true  solution,  if  properly  prepared,  appears 
optically  void.  With  the  ultramicroscope  it  is  possible  to  count 
the  number  of  particles  present  in  a  given  volume  of  a  col- 
loidal solution.  By  means  of  a  chemical  analysis,  the  mass  of 
colloid  per  unit  of  volume  can  be  determined  and  from  this  the 
average  mass  of  each  particle  can  be  calculated.  If  the  particles 
be  assumed  to  be  spherical  in  shape  and  to  have  the  same  density 
as  larger  masses  of  the  same  substance,  we  can  calculate  the 
volume  of  a  single  particle  and  from  this  its  diameter.  Thus, 
Burton  *  in  his  experiments  on  gold,  silver  and  platinum  sols,  found 
the  average  diameter  of  the  colloidal  particles  to  range  from  0.2  to 
0.6  micron. 

Zsigmondy's  latest  ultramicroscope,  Fig.  68,  consisting  of  two 
compound  microscopes  placed  at  right  angles  and  haying  their 
objectives  so  cut  away  as  to  permit  them  to  be  brought  together 
in  focus,  enables  the  observer  to  discern  particles  whose  diameters 
range  from  1  to  2  milli-microns. 

The  ultramicroscopic  character  of  emulsoids  is  by  no  means 
sharply  defined,  notwithstanding  the  fact  that  they  exhibit  the 
Tyndall  phenomenon.  It  has  been  suggested  by  Zsigmondy  that 
the  lack  of  sharpness  in  definition  observed  with  emulsoids  is 
probably  due  to  the  relatively  small  difference  between  the  re- 
fractive indices  of  the  disperse  phase  and  the  dispersion  medium. 
Where  the  difference  between  the  refractive  indices  of  the  two 
phases  is  very  great,  as  in  the  case  of  the  metallic  sols,  excellent 
definition  is  obtained.  It  is  of  interest  to  note  that  although  the 
basic  hydroxides  are  apparently  suspensoids,  yet  in  their  ultra- 
microscopic  characteristics  they  closely  resemble  emulsoids. 

Ultrafiltration.  Almost  all  sols  can  be  filtered  through  ordinary 
filter  paper  without  undergoing  more  than  a  slight  change  in  con- 
centration due  to  initial  adsorption.  The  rate  of  filtration  varies 
widely,  depending  upon  the  viscosity  of  the  sol.  As  a  general  rule, 
*  Phil.  Mag.,  ii,  425  (1906)  .• 


COLLOIDS 


243 


emulsoids  filter  more  slowly  than  suspensoids  owing  to  the  high 
viscosity  of  the  former. 


Fig.  68. 

By  filtering  an  arsenious  sulphide  sol  through  a  porous  earthen- 
ware filter,  Linder  and  Picton  *  succeeded  in  obtaining  four  differ- 

*  Jour.  Chem.  Soc.,  61,  148  (1892). 


244  THEORETICAL  CHEMISTRY 

ent  sizes  of  particles  which  they  described  as  follows:  —  (1)  visible 
under  the  microscope,  (2)  exhibited  the  Tyndall  phenomenon,  (3) 
retained  by  porous  plate,  and  (4)  passed  through  porous  plate  un- 
changed. By  employing  plates  of  different  degrees  of  porosity  and 
determining  the  average  size  of  the  pores  which  just  permit  nitra- 
tion, it  is  possible  to  determine  the  size  of  the  particles  which  con- 
stitute the  disperse  phase  of  a  sol.  If  we  make  use  of  a  series  of 
graduated  filters,  prepared  by  impregnating  filter  paper  with  a 
solution  of  collodion  in  acetic  acid,  it  is  not  only  possible  to  sepa- 
rate suspensoids  from  their  dispersion  media,  but  also  to  effect  the 
concentration  of  emulsoids.  Furthermore  such  filters  are  useful 
in  removing  impurities  from  sols,  the  impurity  passing  through  the 
filter  in  a  manner  similar  to  the  passage  of  the  solvent  through  the 
membrane  in  the  process  of  dialysis.  Ultrafiltration  is  an  exceed- 
ingly complex  process  involving  the  phenomena  of  adsorption  and 
dialysis  in  addition  to  the  ordinary  process  of  mechanical  separation. 
The  complexity  of  the  process  is  well  illustrated  by  the  phenomena 
attendant  upon  the  filtration  of  almost  any  positive  hydrosol. 
Thus,  if  we  attempt  to  filter  a  ferric  hydroxide  hydrosol  through  a 
porous  plate,  or  even  through  an  ordinary  filter  paper,  we  shall 
find  that  the  colloid  will  be  partially  retained  by  the  filter.  This  is 
due  to  the  fact  that  the  filter  becomes  negatively  charged  in  con- 
tact with  water  and,  on  the  entrance  of  the  positively  charged  sol 
into  the  pores  of  the  filter,  the  colloid  is  immediately  discharged 
and  the  disperse  phase  precipitated.  After  the  pores  of  the  filter 
become  partially  stopped  with  particles  of  the  colloid,  the  sol  will 
then  pass  through  unchanged. 

Classification  of  Dispersoids.  There  is  abundant  evidence  in 
favor  of  the  view  that  colloidal  solutions  and  simple  suspensions 
are  closely  related.  Suspensions  of  all  grades  exist,  from  those  in 
which  the  suspended  particles  are  coarse-grained  and  visible  to 
the  naked  eye,  down  to  those  in  which  a  high-power  microscope 
is  required  to  render  the  suspended  particles  visible.  Colloidal 
solutions  have  also  been  shown  to  be  non-homogeneous,  the 
presence  of  discrete  particles  being  revealed  by  means  of  the 
ultramicroscope.  It  follows,  therefore,  that  the  size  of  the  particles 
in  solution  determines  whether  a  substance  is  to  be  considered  as 


COLLOIDS 


245 


a  colloid  or  not.  At  one  extreme  we  have  true  solutions  in  which 
no  lack  of  homogeneity  can  be  detected,  even  by  the  ultramicro- 
scope,  and  at  the  other  extreme  we  have  coarse-grained  suspen- 
sions, in  which  the  particles  are  visible  to  the  naked  eye.  Between 
these  two  limits  all  possible  degrees  of  subdivision  are  possible 
and  it  is  a  very  difficult  matter  to  draw  sharp  lines  of  distinction 
between  true  solutions  and  colloidal  solutions  on  the  one  hand, 
and  between  colloidal  solutions  and  suspensions  on  the  other. 
One  of  the  most  satisfactory  schemes  of  classification  is  that  of 
von  Weimarn  and  Wo.  Ostwald.*  Because  of  the  fact  that  sus- 
pensions, colloidal  solutions,  and  true  solutions  represent  varying 
degrees  of  dispersion  of  the  solute,  all  three  types  of  system  are 
termed  by  these  authors,  dispersoids.  The  dispersoids  are  classi- 
fied as  shown  in  the  accompanying  diagram. 

DISPERSOIDS. 


Coarse  dispersions  (sus- 

pensions,    emulsions, 

etc.). 

Magnitude  of  particles 

greater  than  0.1  M-* 

Colloidal  solutions.           Molecular            Ionic 

Magnitude  of  particles       dispersoids.     dispersoids. 

between  0.1  M  and  1  MM- 

Decreasing  degree  of  Magnitude  of  particles, 

"colloidity."  about  1  MM  and  smaller. 

Increasing  degree  of  dis- 
persion. 

*  1  M  =  1  micron  =  0.001  mm. 

Density  of  Colloidal  Solutions.  As  we  have  seen,  suspensoids 
are  commonly  regarded  as  sols  in  which  the  disperse  phase  is  solid, 
while  emulsoids  are  considered  to  be  sols  in  which  the  disperse 
phase  is  liquid.  While  this  distinction  between  the  two  classes  of 
sols  is  generally  well  defined,  it  should  be  borne  in  mind  that  there 
*  Koll.  Zeitschrift,  3,  26  (1908). 


246 


THEORETICAL  CHEMISTRY 


are  colloidal  solutions  in  which  it  is  extremely  difficult  to  determine 
the  physical  state  of  the  disperse  phase. 

The  fundamental  difference  between  suspensoids  and  emulsoids 
manifests  itself  most  clearly  in  those  properties  which  undergo 
appreciable  change  in  consequence  of  solution.  Among  these  may 
be  mentioned  density,  viscosity  and  surface  tension. 

It  was  shown  by  Linder  and  Picton  *  that  the  density  of 
suspensoids  follows  the  law  of  mixtures.  This  is  clearly  shown 
by  the  following  table  in  which  are  given  the  observed  and 
calculated  values  of  the  density  of  a  series  of  arsenious  sulphide 
sols. 

DENSITY  OF  ARSENIOUS  SULPHIDE  SOLS. 


As2S3  (grams  per 
liter). 

Density  (obs.). 

Density  (calc.). 

44 

1.033810 

1.033810 

22 

1.016880 

1.016905 

11 

1.008435 

1.008440 

2.45 

1.002110 

1.002100 

0.1719 

1.000137 

1.000134 

The  density  of  emulsoids,  on  the  other  hand,  cannot  be  calcu- 
lated from  the  composition  of  the  sol.  This  fact  may  be  taken  as 
evidence  in  favor  of  the  view  that  a  closer  relation  exists  between 
the  disperse  phase  and  the  dispersion  medium  in  emulsoids  than  in 
suspensoids. 

Viscosity  of  Colloidal  Solutions.  Owing  to  the  fact  that  the 
concentrations  of  most  suspensoids  are  relatively  small,  it  follows 
that  their  viscosities  differ  but  little  from  the  viscosity  of  the  pure 
dispersion  medium.  In  general,  it  may  be  said  that  the  viscosity 
of  suspensoid  sols  is  slightly  greater  than  that  of  the  dispersion 
medium. 

On  the  other  hand,  the  viscosity  of  emulsoid  sols  is  frequently 
much  greater  than  that  of  the  pure  dispersion  medium.     The  vis- 
cosity of  emulsoids  also  increases  with  increasing  concentration, 
as  is  shown  by  the  data  of  the  following  table. 
*  Jour.  Chem.  Soc.,  67,  71  (1895). 


COLLOIDS 


247 


VISCOSITY  OF  EMULSOIDS. 


Sol. 

Temp.  20° 
Concentration. 

Viscosity. 

Per  cent. 

Gelatine 

1 

0  021 

Gelatine 

2 

0  037 

Silicic  acid.  .  .  . 

0.81 

0.012 

Silicic  acid.  . 

0  99 

0.016 

Silicic  acid.  .  . 

1.96 

0.032 

Silicic  acid  — 

3.67 

0.165 

Viscosity  of  water  at  20°  =  0.0120. 

Surface  Tension  of  Colloidal  Solutions.  The  surface  tension 
of  suspensoid  sols  has  been  shown  by  Linder  and  Picton  to  be 
practically  identical  with  that  of  the  dispersion  medium. 

As  a  general  rule,  the  surface  tension  of  emulsoid  sols  is  appre- 
ciably smaller  than  that  of  the  dispersion  medium.  According  to 
Quincke  *  the  surface  tensions  of  gelatine  sols  are  appreciably  less 
than  the  surface  tension  of  the  dispersion  medium.  The  differ- 
ence between  suspensoids  and  emulsoids  in  respect  to  surface 
tension  undoubtedly  accounts  for  the  fact  that,  in  general,  the 
former  are  not  adsorbed  while  the  latter  are. 

Osmotic  Pressure  of  Colloidal  Solutions.  The  osmotic  pres- 
sure of  colloidal  solutions  is  very  small.  This  is  what  we  should 
expect  with  solutions  of  substances  which  exhibit  a  slow  rate  of 
diffusion.  As  has  been  pointed  out,  diffusion  is  closely  connected 
with  osmotic  pressure;  hence,  if  the  rate  of  diffusion  is  slow,  the 
osmotic  pressure  exerted  by  the  solution  should  be  small.  In 
some  eases  the  osmotic  pressure  is  so  small  as  to  escape  detection. 
The  experimental  determination  of  the  osmotic  pressure  of  col- 
loidal solutions  is  complicated  by  the  difficulty  of  removing  the 
last  traces  of  electrolytes  from  the  colloid.  Owing  to  their  great 
osmotic  activity,  the  presence  of  the  merest  trace  of  electrolytes 
may  mask  the  true  osmotic  effect  of  the  colloid.  It  should  be  borne 
in  mind,  however,  that  semipermeable  membranes  are  much  less 
permeable  by  colloids  than  by  electrolytes  and,  in  consequence  of 
this  fact,  the  impurities  in  the  colloid  would  be  gradually  removed 
by  prolonged  dialysis.  If  the  total  osmotic  pressure  were  due  to 
*  Wied.  Ann.,  Ill,  35,  582  (1885). 


248 


THEORETICAL  CHEMISTRY 


the  presence  of  small  amounts  of  impurities  in  the  colloid,  then  as 
these  are  removed,  the  pressure  should  steadily  diminish  and  ulti- 
mately become  zero.  As  a  matter  of  fact,  the  final  value  of  the 
osmotic  pressure  of  a  colloidal  solution,  although  generally  very 
small  is  never  zero.  This  final,  positive  value  of  the  osmotic 
pressure  has  been  shown  to  be  wholly  independent  of  the  method 
of  preparation  of  the  sol.  Although  differences  in  the  method  of 
preparation  may  introduce  different  impurities  which  give  rise  to 
different  initial  values  of  the  osmotic  pressure,  in  each  case  the 
same  final  value  is  obtained.  Of  course  it  must  be  admitted  that 
the  possibility  exists  that  a  minute  portion  of  electrolyte  which 
cannot  be  removed  by  dialysis  is  retained  by  the  colloid,  but  even 
then  it  is  difficult  to  account  for  the  constancy  of  the  final  value  of 
the  osmotic  pressure  irrespective  of  the  method  of  preparation  of 
the  sol.  The  values  of  the  osmotic  pressure  of  suspenoids  are  in- 
variably small  and  by  no  means  concordant. 

The  following  table  gives  the  results  obtained  by  Duclaux  *  with 
colloidal  solutions  of  ferric  hydroxide. 

OSMOTIC  PRESSURE  OF  COLLOIDAL  FERRIC  HYDROXIDE. 


Cone.  Fe(OH)3 
Per  cent. 

Pressure  in  cm. 
of  Water. 

1.08 

0.8 

2.04 

2.8 

3.05 

5.6 

5.35 

12.5 

8.86 

22.6 

Inspection  of  the  table  shows  that,  even  in  the  most  concen- 
trated solution,  the  osmotic  pressure  is  very  small.  Furthermore, 
it  is  apparent  that  although  the  osmotic  pressure  increases  with 
the  concentration,  the  variables  are  not  proportional.  Observa- 
tions on  the  variation  of  the  osmotic  pressure  of  colloidal  solu- 
tions with  temperature,  show  that,  in  general,  as  the  temperature 
is  raised  the  pressure  increases  at  a  more  rapid  rate  than  that 
required  by  the  law  of  Gay-Lussac. 

Employing  membranes  of  collodion  and  parchment  paper, 
*  Compt.  rend.,  140,  1544  (1905);  Jour.  Chim.  Phys.,  7,  405  (1909). 


COLLOIDS  249 

Lillie  *  and  others  have  demonstrated  that  'the  values  of  the  os- 
motic pressure  of  emulsoids  are,  in  general,  considerably  greater 
than  the  corresponding  values  obtained  with  suspensoids.  The 
osmotic  pressures  of  several  typical  emulsoids  are  given  in  the 
following  table. 

OSMOTIC  PRESSURES  OF  EMULSOIDS. 


Sol. 

Concentration 
(grams  per  liter)  . 

Osmotic  Pressure 
(mm.  of  mercury). 

Egg  albumin  

12.5 

20 

Gelatine 

12  5 

6 

Starch  iodide 

30 

15 

Dextrin 

10 

165 

It  will  be  observed  that  the  values  of  the  osmotic  pressure  in  the 
preceding  table  are  appreciably  greater  than  those  given  for  ferric 
hydroxide  hydrosols.  This  is  in  agreement  with  the  well-estab- 
lished fact  that  emulsoids  diffuse  more  rapidly  than  suspensoids. 
It  has  been  observed,  that  the  value  of  the  osmotic  pressure  of 
gelatine  solutions  at  ordinary  temperatures  can  be  increased  by 
maintaining  the  solutions  at  a  higher  temperature  for  a  short  time 
and  then  cooling  to  the  initial  temperature.  After  standing  for 
several  days  at  the  original  temperature,  however,  the  osmotic 
pressure  of  the  solution  returns  to  its  former  value.  This  phenom- 
enon would  seem  to  indicate  th^t  the  osmotic  pressure  of  colloidal 
solutions  is  not  completely  defined  by  the  two  variables,  tempera- 
ture and  concentration.  It  has  been  suggested  that  the  degree  of 
aggregation  of  the  colloid  is  partially  dependent  upon  the  tempera- 
ture; the  molecular  aggregates  tending  to  break  up  as  the  tempera- 
ture is  raised,  thus  increasing  the  number  of  dissolved  units  and 
therefore  causing  a  corresponding  increase  in  the  osmotic  pressure. 

Molecular  Weight  of  Colloids.  We  have  already  learned  that 
the  knowledge  of  the  osmotic  pressure  of  a  solution  enables  us  to 
calculate  the  molecular  weight  of  the  solute,  provided  the  solu- 
tion is  dilute  and  obeys  the  gas  laws.  As  we  have  seen,  other 

*  Am.  Jour.  Physiol.,  20,  127  (1907), 


250  THEORETICAL  CHEMISTRY 

factors  than  concentration  and  temperature  determine  the  osmotic 
pressure  of  colloidal  solutions,  so  that  we  are  not  justified  in 
attempting  to  calculate  the  molecular  weight  of  a  colloid  from  the 
observed  value  of  the  osmotic  pressure  of  its  solution.  Values  for 
the  molecular  weight  of  colloids  calculated  from  their  effect  on  the 
vapor  pressure,  the  boiling-point,  and  the  freezing-point  of  the 
solvent  are  also  untrustworthy,  since  the  same  factors  which 
influence  the  osmotic  pressure  necessarily  affect  these  related 
properties.  This  becomes  evident  when  we  reflect  that  an  osmotic 
pressure  of  1  mm.  of  mercury  corresponds  to  a  depression  of  the 
freezing-point  of  about  0°.0001.  Owing  to  the  difficulty  of  ob- 
taining absolutely  pure  emulsoid  sols,  all  determinations  of  their 
freezing-point  depressions  must  be  affected  with  an  experimental 
error  appreciably  larger  than  the  observed  depression.  Hydrosols 
of  albumin,  gelatine,  etc.,  prepared  with  extreme  care  by  Bruni 
and  Pappada  *  failed  to  produce  any  detectable  depression  of  the 
freezing-point  of  water. 

Electroendosmosis.  The  movement  of  a  liquid  through  a 
porous  diaphragm,  due  to  the  passage  of  an  electric  current  be- 
tween two  electrodes  placed  on  opposite  sides  of  the  diaphragm,  is 
known  as  eledroendosmosis.  This  phenomenon,  which  was  first 
observed  by  Reuss  in  1807,  has  since  been  made  the  subject  of 
numerous  investigations  by  Wiedemann,f  Quincke  J  and  Perrin,§ 
the  latter  having  worked  out  a  satisfactory  theoretical  interpreta- 
tion of  the  phenomenon.  If  a  porous  partition  be  placed  in  the 
horizontal  portion  of  a  U-tube  and  an  electrode  be  inserted  in  each 
arm  of  the  tube,  it  will  be  found,  on  filling  the  tube  with  a  feebly 
conducting  liquid  and  passing  a  current,  that  the  liquid  will  com- 
mence to  rise  in  one  arm  of  the  tube  and  will  continue  to  rise  until 
a  definite  equilibrium  is  established.  For  a  given  difference  of 
potential  between  the  two  electrodes,  there  will  be  a  definite  differ- 
ence in  the  level  of  the  liquid  in  the  two  arms  of  the  tube.  The 
majority  of  substances  acquire  a  negative  electric  charge  when 

*  Rend.  R.  Accad.  dei  Lincei,  (5),  9,  354  (1900). 

t  Pogg.  Ann.,  87,  321  (1852). 

t  Ibid.,  113,  513  (1861). 

§  Jour.  Chim.  Phys.,  2,  601  (1904). 


COLLOIDS  251 

immersed  in  water.  The  water,  under  these  conditions,  becomes 
positively  charged  and  will,  in  consequence,  migrate  toward  the 
cathode.  On  the  other  hand,  certain  substances  acquire  a  positive 
charge  on  immersion  in  water  and  in  these  cases  the  direction  of 
migration  will  obviously  be  reversed. 

It  has  been  found  that  acids  cause  negative  diaphragms  to  be- 
come less  negative  and  positive  diaphragms  to  become  more  posi- 
tive. The  action  of  alkalies  is,  as  we  should  expect,  the  reverse  of 
that  of  acids.  There  is  an  interesting  connection  between  the 
valence  of  the  ions  resulting  from  the  dissociation  of  dissolved 
salts,  and  the  difference  of  potential  existing  between  the  liquid 
and  the  diaphragm.  When  the  diaphragm  is  positively  charged, 
the  difference  of  potential  is  found  to  be  'conditioned  by  the 
valence  of  the  anion,  and  when  the  diaphragm  is  negatively 
charged,  the  difference  of  potential  is  determined  by  the  valence 
of  the  cation. 

Cataphoresis.  When  a  difference  of  potential  is  established 
between  two  electrodes  immersed  in  a  suspension  of  finely-divided 
quartz  or  shellac,  the  suspended  particles  move  toward  the  positive 
electrode.  This  phenomenon  is  called  cataphoresis  and  was  first 
observed  by  Linder  and  Picton.*  They  showed  that  when  the 
terminals  of  an  electric  battery  are  connected  to  two  platinum 
electrodes  dipping  into  a  colloidal  solution  of  arsenious  sulphide, 
there  is  a  gradual  migration  of  the  colloid  to  the  positive  pole.  A 
similar  experiment  with  a  solution  of  colloidal  ferric  hydroxide 
resulted  in  the  transport  of  the  dissolved  colloid  to  the  negative 
pole.  It  follows,  therefore,  that  the  particles  of  colloidal  arsenious 
sulphide  are  negatively  charged,  while  those  of  colloidal  ferric 
hydroxide  carry  a  positive  charge.  It  has  been  found  that  all 
colloids  carry  an  electric  charge.  In  the  table  on  page  252,  some 
typical  colloids  are  classified  according  to  the  character  of  their 
electrification  in  aqueous  solution. 

The  nature  of  the  charge  varies  with  the  dispersion  medium  used, 
colloidal  solutions  in  turpentine,  for  example,  having  charges  oppo- 
site to  those  in  water. 

Direct  measurements  of  the  velocity  with  which  the  particles 
*  Jour.  Chem.  Soc.,  61,  148  (1892). 


252 


THEORETICAL  CHEMISTRY 


move  in  cataphoresis  have  been  made  by  Cotton  and  Mouton.* 
By  observing  with  a  microscope  the  distance  over  which  a  single 

ELECTRICAL  CHARGES  OF  HYDROSOLS. 


Electro-positive. 

Electro-negative. 

Metallic  hydroxides 
Methyl  violet 
Methylene  blue 
Magdala  red 
Bismarck  brown 
Haemoglobin 

All  the  metals 
Metallic  sulphides 
Aniline  blue 
Indigo 
Eosine 
Starch 

particle  traveled  in  a  given  interval  of  time  under  a  definite  po- 
tential gradient,  they  calculated  the  average  velocity  of  migration 
of  a  number  of  suspensoids.  The  following  table  gives  the  velocity 
of  migration  of  a  few  typical  suspensoids. 

VELOCITY  OF  MIGRATION  OF  SUSPENSOIDS.  f 


Suspensoid. 

Average  Diameter 
of  Particles. 

Velocity    cm.  /sec.    for 
Unit  Potential 
Gradient. 

Arsenic  trisulphide.  ...        ...    . 

50M 

22X10"5 

Quartz  

1   MM 

30X10-5 

Gold  (colloidal)  

<  100  MM 

40X10-5 

Platinum  (colloidal) 

<100  MM 

30X1Q-* 

Silver  (colloidal) 

<  100  MM 

23  6X10-5 

Bismuth  (colloidal)  .  . 

<  100  MM 

11  OX10-5 

Lead  (colloidal)  

<100  uu 

12  OX1Q-* 

Iron  (colloidal)  

<100  U.LL 

19  OxlO~6 

Ferric  hydroxide  (colloidal)  

100  MM 

30  OX  10-* 

t  Freundlich,  Kapillarchemie,  p.  234. 

It  will  be  observed  that  not  only  are  the  velocities  of  migration 
nearly  constant,  but  also  that  they  are  apparently  independent  of 
the  size  and  nature  of  the  particles. 

The  presence  of  electrolytes,  especially  acids  and  bases,  exercises 
a  marked  effect  upon  the  electrical  behavior  of  suspensoids. 
Owing  to  the  comparative  instability  of  suspensoids,  the  addition 
of  electrolytes  usually  results  in  the  complete  precipitation  of  the 
colloid. 

*  Jour.  Chim.  Phys.,  4,  365  (1906). 


COLLOIDS  253 

Emulsoids  also  show  the  phenomenon  of  cataphoresis,  but  their 
velocities  of  migration  are  appreciably  less  than  the  corresponding 
velocities  of  suspensoids,  and  their  behavior  in  an  electric  field  is 
such  as  to  make  it  appear  quite  probable  that  the  character  of  their 
electric  charge  is  entirely  fortuitous.  Furthermore,  emulsoids  are 
much  more  susceptible  to  the  influence  of  electrolytes  than  are 
suspensoids. 

W.  B.  Hardy  *  has  found  that  the  direction  of  migration  of 
albumin,  modified  by  heating  to  100°  C.,  is  dependent  upon  the 
reaction  of  the  dispersion  medium.  A  very  small  quantity  of  free 
base  causes  the  particles  of  albumin  to  move  toward  the  positive 
electrode,  while  the  addition  of  an  equally  small  amount  of  acid 
results  in  a  reversal  of  the  direction  of  migration.  Similar  reversals 
of  charge  have  been  observed  by  Burton  f  in  colloidal  solutions  of 
gold  and  silver.  When  small  amounts  of  aluminium  sulphate  are 
added  to  colloidal  solutions  of  these  metals,  the  charge  is  gradually 
neutralized  and  eventually  the  colloidal  particles  acquire  a  reversed 
charge. 

Electrical  Conductance  of  Colloidal  Solutions.  The  electrical 
conductance  of  suspensoids  differs  so  slightly  from  that  of  the  pure 
dispersion  medium  that  it  is  difficult  to  decide  whether  the  small 
increase  in  conductance  may  not  be  due  to  the  presence  of  traces 
of  adsorbed  electrolytes.  In  order  to  ascertain  to  what  extent  the 
conductance  of  suspensoids  is  dependent  upon  the  presence  of 
adsorbed  electrolytes,  Whitney  and  Blake  J  investigated  the  effect 
of  successive  electrolyses  upon  the  conductance  of  a  gold  hydrosol. 
If  the  pure  sol  is  incapable  of  enhancing  the  conductance  of  the 
dispersion  medium,  then  as  the  sol  is  subjected  to  successive  elec- 
trolyses, the  conductance  should  steadily  diminish  and  ultimately 
become  identical  with  that  of  the  dispersion  medium.  Whitney 
and  Blake  found  that  the  conductance  converged  to  a  definite 
limiting  value  which  was  slightly  greater  than  the  conductance  of 
the  dispersion  medium.  From  these  experiments  we  seem  to  be 
warranted  in  concluding  that  suspensoids  conduct  the  electric 
current  very  feebly. 

*  Jour.  Physiol.,  24,  288  (1899). 

t  Phil.  Mag.,  12,  472  (1906). 

J  Jour.  Am.  Chem.  Soc.,  26,  1339  (1904). 


254  THEORETICAL   CHEMISTRY 

Emulsoids  appear  to  conduct  rather  better  than  suspensoids. 
Whitney  and  Blake  measured  the  conductance  of  silicic  acid  and 
gelatine  sols  and  found  the  specific  conductance  of  the  former  to  be 
100  X  10~6  reciprocal  ohms  and  that  of  the  latter  to  be  68  X  10~6 
reciprocal  ohms.  On  the  other  hand,  Pauli  *  found  that  an  albumin 
sol  which  had  been  prepared  with  extreme  care  was  virtually  a 
non-conductor.  It  should  be  remembered,  however,  that  the 
albumins  are  closely  related  to  the  simple  amino-acids  which  are 
known  to  be  exceedingly  poor  conductors. 

Precipitation  of  Colloids  by  Electrolytes.  One  of  the  most 
important  and  interesting  divisions  of  the  chemistry  of  colloids  is 
that  which  treats  of  the  precipitation  of  suspensoids  and  emulsoids 
by  electrolytes.  In  general,xit  may  be  said  that  the  precipitation 
of  colloids  by  electrolytes  is  an  irreversible  process.  Colloidal 
solutions  are  more  or  less  unstable  systems  irrespective  of  the 
methods  employed  in  their  preparation,  and  the  addition  of  a  small 
amount  of  an  electrolyte  is  usually  found  to  be  sufficient  to  cause 
the  sol  immediately  to  become  opalescent,  and  ultimately  to  pre- 
cipitate, leaving  the  dispersion  medium  perfectly  clear  and  free 
from  the  disperse  phase.  Some  exceptions  to  this  general  state- 
ment as  to  the  behavior  of  colloids  are  known.  For  example, 
Whitney  and  Blake  f  found  that  precipitated  gold  could  be  caused 
to  return  to  the  colloidal  state  by  treatment  with  ammonia,  while 
Linder  and  Picton  {  discovered  that  a  ferric  hydroxide  hydrosol, 
which  had  been  precipitated  with  sodium  chloride,  could  be  re- 
stored to  the  colloidal  condition  by  simply  removing  the  electrolyte 
with  water.  The  sedimentation  of  suspensions,  such  as  kaolin  in 
water,  is  also  promoted  by  the  addition  of  electrolytes. 

On  the  other  hand,  non-electrolytes  have  no  effect  upon  the 
stability  of  suspensoids  or  simple  suspensions. 

Precipitation  of  Suspensoids.  The  phenomenon  of  the  pre- 
cipitation of  suspensoids  has  been  carefully  investigated  by 
Freundlich.§  He  has  found  that  an  amount  of  electrolyte  which 

*  Beitrag.  Chem.  Phys.  Path.,  7,  531  (1906). 

t  Jour.  Am.  Chem.  Soc.,  26,  1341  (1904). 

t  Jour.  Chem.  Soc.,  61,  114  (1892);  87,  1924  (1905). 

§  Zeit.  phys.  Chem.,  44,  131  (1903). 


COLLOIDS 


255 


is  incapable  of  bringing  about  an  instantaneous  precipitation,  may 
become  effective  after  an  interval  of  time.  He  has  also  shown 
that  the  total  quantity  of  electrolyte  required  to  precipitate  a 
suspensoid  completely  depends  upon  whether  the  electrolyte  is 
added  all  at  one  time  or  in  successive  portions.  In  order  to  com- 
pare the  precipitating  action  of  various  electrolytes,  Freundlich 
proposed  the  following  procedure,  which  prevents  the  possibility 
of  irregularities  due  to  the  time  factor:  —  To  20  cc.  of  a  solution 
of  a  suspensoid,  2  cc.  of  the  solution  of  the  electrolyte  are  added, 
the  resulting  solution  being  shaken  vigorously;  the  mixture  is  then 
set  aside  for  two  hours,  after  which  a  small  portion  is  filtered  off, 
and  the  filtrate  is  examined  for  the  suspensoid.  In  the  following 
table  some  of  the  results  obtained  by  Freundlich  with  colloidal  solu- 
tions of  ferric  hydroxide  are  given.  The  data  represent  the  mini- 
mum concentration  for  each  electrolyte  which  produced  precipita- 
tion in  two  hours. 

It  will  be  seen  that  very  small  amounts  of  the  electrolytes  are 
required  to  precipitate  the  suspensoid,  and  further,  that  the  pre- 
cipitating power  of  an  electrolyte  is  dependent  upon  the  charge  of 
the  negative  ion.  The  greater  the  charge,  the  smaller  is  the 
quantity  of  electrolyte  required  to  produce  precipitation. 

PRECIPITATING  ACTION  OF  ELECTROLYTES  ON  FERRIC 
HYDROXIDE  HYDROSOL. 

(16  milli-mols  Fe(OH)2  per  liter.) 


Electrolyte. 

Concentration 
(milli-mols  per 
liter). 

NaCl 

9.25 

KC1 

9.03 

BaCl2 

9.64 

KN03 

11.9 

KBr 

12.5 

Ba(N03)2 

14.0 

KI 

16.2 

HC1 

400  ca. 

Ba(OH)2 

0.42 

H3(SO)4 

0.204 

MgS04 

0.217 

K2Cr2O7 

0.194 

H2SO4 

0.5ca. 

256 


THEORETICAL  CHEMISTRY 


The  significance  of  the  relation  between  ionic  charge  and  pre- 
cipitating power  was  first  pointed  out  by  Hardy,*  who  formulated 
the  following  rule :  —  The  precipitation  of  a  colloidal  solution  is  de- 
termined by  that  ion  of  an  added  electrolyte  which  has  an  electric  charge 
opposite  in  sign  to  that  of  the  colloidal  particles. 

It  has  already  been  pointed  out  that  colloidal  particles  of  arse- 
nious  sulphide  are  negatively  charged,  hence,  according  to  Hardy's 
rule,  the  positive  ions  of  the  added  electrolyte  will  condition  the 
precipitation  of  the  suspensoid.  The  experiments  of  Freundlich 
confirm  this  prediction,  as  is  shown  by  the  following  table: 

PRECIPITATING  ACTION  OF  ELECTROLYTES  ON  ARSENIOUS 
SULPHIDE  HYDROSOL. 

.     (7.54  milli-mols  As2S3  per  liter.) 


t  Electrolyte. 

Concentration 
(milli-mols  per 
liter). 

KC1 

49.5 

KNO3 

50.0 

KC2H3O2 

110.0 

NaCl 

51.0 

LiCl 

58.4 

MgCl2 

0.717 

MgS04 

0.810 

.    CaCl2 

0.649 

SrCl2 

0.635 

BaCl2 

0.691 

Ba(N03)2 

0.687 

ZnCl2 

0.685 

Aids 

0.093 

A1(N03)3 

0.095 

Precipitation  and  Valence.  An  examination  of  the  preceding 
tables  reveals  the  fact  that  although  the  ionic  concentration  neces- 
sary to  bring  about  precipitation,  in  accordance  with  Hardy's  rule, 
decreases  with  increasing  valence,  the  diminution  in  concentration 
is  not,  as  we  might  expect,  inversely  proportional  to  the  valence  of 
the  precipitating  ion.  The  absence  of  any  simple  quantitative 
relation  between  the  valence  of  an  ion  and  its  precipitating  con- 

*  Zeit.  phys.  Chem.,  33,  385  (1900). 


COLLOIDS  257 

centration  is  undoubtedly  due  to  the  influence  of  several  potent 
factors,  such  as  adsorption  and  the  protective  action  of  ions  whose 
electric  charge  is  of  the  same  sign  as  that  of  the  colloidal  substance. 

Precipitation  of  Emulsoids.  The  action  of  electrolytes  on 
emulsoids  is  much  more  obscure  than  the  action  of  electrolytes  on 
suspensoids.  Nothing  approaching  a  generalization  similar  to 
Hardy's  rule  for  the  precipitation  of  suspensoids  has  been  found  to 
apply  to  the  precipitation  phenomena  manifested  by  emulsoids. 
Owing  to  the  fact  that  emulsoids  are  liquids,  and  in  consequence 
of  their  greater  degree  of  dispersity,  it  has  been  suggested  that 
emulsoids  probably  resemble  true  solutions  more  closely  than  sus- 
pensoids. In  fact  there  is  reason  for  assuming  that  a  portion  of 
the  colloid  is  actually  dissolved  in  the  dispersion  medium.  This 
may  account  for  the  fact  that  the  precipitation  of  emulsoids  is 
sometimes  reversible  and  sometimes  irreversible.  It  is  to  be  re- 
gretted that,  up  to  the  present  time,  so  many  of  the  investigations 
on  emulsoids  have  been  carried  out  with  materials  of  questionable 
purity  and  of  insufficient  uniformity. 

The  addition  of  salts  to  gelatine  generally  causes  irreversible 
precipitation,  provided  the  concentration  of  the  salt  is  not  too  low. 

The  precipitation  of  albumin  by  some  salts  is  reversible  while 
by  others  it  is  irreversible.  Those  transformations  which  are 
initially  reversible  gradually  become  irreversible  on  standing. 
Although  relatively  concentrated  solutions  of  salts  of  the  alkalies 
and  alkaline  earths  are  required  to  precipitate  albumin,  very  dilute 
solutions  of  the  salts  of  the  heavy  metals  are  found  to  be  sufficient 
to  bring  about  complete  precipitation. 

Action  of  Heat  on  Emulsoids.  When  an  albumin  hydrosol  is 
gradually  heated,  a  temperature  is  ultimately  reached  at  which 
coagulation  occurs.  The  exact  nature  of  this  transformation  is 
not  understood,  but  it  is  believed  to  be  largely  chemical.  This 
belief  is  based  upon  the  fact  that  the  reaction  of  the  natural  albu- 
mins toward  litmus  is  altered  by  heating.  Slight  acidity  of  the  sol 
is  essential  to  complete  coagulation,  while  an  excess  or  a  de- 
ficiency of  acids  causes  a  portion  of  the  albumin  to  remain  in  the 
sol.  The  presence  of  various  salts  has  been  found  to  exert  a 
marked  influence  on  the  temperature  of  coagulation  of  albumin. 


258  THEORETICAL  CHEMISTRY 

The  coagulation  temperature  is  invariably  raised  at  first,  attaining 
a  constant  value  in  some  cases,  while  in  others  it  decreases  after 
reaching  a  maximum  temperature.  It  is  especially  interesting  to 
note  that  the  anions  of  the  salts  follow  the  usual  lytropic  sequence. 
The  effect  of  heat  on  a  gelatine  sol  is  very  different  from  its 
effect  on  an  albumin  sol.  If  a  fairly  concentrated  gelatine  sol  is 
heated  and  then  permitted  to  cool,  it  sets  into  a  jelly  which  is  not 
reconverted  into  a  sol  when  the  temperature  is  again  raised.  Fur- 
thermore, the  change  does  not  take  place  at  a  definite  temperature. 
In  studying  the  phenomenon  of  gelation,  either  the  temperature 
or  the  time  of  gelation  may  be  determined.  The  melting-point  of 
pure  gelatine  ranges  from  26°  to  29°  while  the  solidifying  tempera- 
ture lies  between  25°  and  18°.  The  melting  and  solidifying  tem- 
perature of  gelatine  sols  vary  with  the  concentration;  a  5  per  cent 
sol  melts  at  26°.  1  and  solidifies  at  17°.8,  while  a  15  per  cent  sol 
melts  at  29°.4  and  solidifies  at  25°.5.  The  temperature  of  the  gel- 
sol  transformation  is  affected  by  the  addition  of  salts,  some  tending 
to  raise  the  temperature  of  gelation  and  others  to  lower  it.  The 
order  of  the  anions  arranged  according  to  their  influence  on  the 
gel-sol  transformation  is  as  follows :  — 

Raising  temperature :  SO4  >  Citrate  >  Tartrate  >  Acetate  (H20). 
Lowering  temperature:  (H20)  Cl  <  C103  <  NO3  <  Br  <  I. 

The  same  lytropic  order  was  found  by  Schroeder  *  in  an  investi- 
gation of  the  viscosity  of  gelatine  sols. 

It  is  noteworthy  that  the  influence  of  salts  on  the  temperature 
of  gelation  of  agar-agar  and  other  similar  substances  is  analogous 
to  their  effect  on  gelatine,  the  same  lytropic  sequence  being  main- 
tained. 

Protective  Colloids.  The  precipitating  action  of  electrolytes 
on  suspensoids  may  be  inhibited  by  adding  to  the  solution  of  the 
suspensoid  a  reversible  colloid.  The  protective  action  of  a  re- 
versible colloid  is  not  due,  as  might  be  supposed,  to  the  increased 
viscosity  of  the  medium  and  the  resultant  resistance  to  sedimenta- 
tion, since  amounts  of  a  reversible  colloid,  too  minute  to  produce 
any  appreciable  increase  in  the  viscosity  of  the  medium,  can  pre- 

*  Zeit.  phys.  Chem.,  45,  75  (1903). 


COLLOIDS 


259 


vent  precipitation.  Thus,  Bechhold  *  has  shown  that  while  a 
mixture  of  1  cc.  of  a  suspension  of  mastic  and  1  cc.  of  a  0.1  molar 
solution  of  MgS04  diluted  to  3  cc.  with  water,  is  completely  pre- 
cipitated in  15  minutes,  no  precipitation  will  occur  within  24  hours, 
if  two  drops  of  a  1  per  cent  solution  of  gelatine  be  added  before 
diluting  to  3  cc. 

Gum  arabic  and  ox-blood  serum  exert  a  similar  protective  action 
when  added  to  a  suspension  of  mastic.  The  protective  power  of 
reversible  colloids  differs  widely  and  Zsigmondy  f  has  attempted 
to  make  this  the  basis  of  a  method  of  classification  of  colloidal  sub- 
stances. A  red  solution  of  colloidal  gold  becomes  blue  on  the 
addition  of  a  small  amount  of  sodium  chloride,  owing  to  the  in- 
crease in  the  size  of  the  colloidal  particles.  Various  colloidal  sub- 
stances when  added  to  a  red  colored  gold  sol  protect  the  colloidal 
particles  from  precipitation  by  a  solution  of  sodium  chloride,  no 
change  in  color  following  the  addition  of  the  electrolyte.  A  definite 
amount  of  each  colloidal  substance  is  required  to  prevent  the 
change  from  red  to  blue  in  the  color  of  the  gold  sol.  In  employing 
this  color  change  as  a  means  of  differentiating  colloidal  substances, 
Zsigmondy  introduced  the  "gold  number,"  which  may  be  defined 
as  the  weight  in  milligrams  of  a  colloidal  substance  which  is  just 
insufficient  to  prevent  the  change  from  red  to  blue  in  10  cc.  of  a 
gold  sol  after  the  addition  of  1  cc.  of  a  10  per  cent  solution  of 
sodium  chloride.  The  following  table  gives  the  gold  numbers  of  a 

few  colloids. 

GOLD  NUMBERS  OF  COLLOIDS. 


Colloid. 

Gold  Number. 

Gelatine 

0  005-0  01 

Casein  (in  ammonia) 

0  01 

Egg-albumin  
Gum  arabic  
•Dextrin  
Starch,  wheat  
Starch,  potato 

0.15-0.25 
0.15-0.25)0.5-4 
6-12:  10-20 
4-6  (about) 
25  (about) 

Sodium  stearate  .  . 

10  (at  60  °)-  0  01  (at 

100°) 

Sodium  oleate  
Cane  sugar  
Urea  

0.4-1 

8 
8 

*  Zeit.  phys.  Chem.,  48,  408  (1904). 
f  Zeit.  analyt.  Chem.,  40,  697  (1901). 


260 


THEORETICAL  CHEMISTRY 


The  gold  number  has  proven  useful  in  differentiating  the  various 
kinds  of  albumin,  as  is  shown  in  the  following  table. 

GOLD  NUMBERS  OF  ALBUMINS. 


Albumin. 

Gold  Number. 

Egg  white  (fresh)  
Globulin 

0.08 

0.02-0.05 

Ovomucoid                       .  •  • 

0.04-0.08 

Albumin  (Merck)  

0.1-0.3 

Albumin  (cryst.)  
Albumin  (alkaline)  

2-8 
0.006-0.04 

The  addition  of  alkali  to  any  one  of  the  first  five  albumins  of  the 
above  table  reduces  the  gold  number  to  that  of  alkaline  albumin. 
Sulphide  sols  may  be  protected  as  well  as  metallic  sols,  and  further- 
more the  ability  to  exert  protective  action  is  not  confined  to  organic 
colloids  alone. 

In  general,  it  may  be  said  that  when  a  suspensoid  sol  is  mixed 
with  an  emulsoid  sol  in  the  proper  proportions,  the  suspensoid  sol 
acquires  most  of  the  characteristic  properties  of  the  protecting 
colloid.  The  masking  of  the  properties  of  a  suspensoid  sol  by  a 
protecting  colloid  is  probably  to  be  ascribed  to  the  formation  of  a 
thin  film  of  adsorbed  emulsoid  over  the  suspensoid. 

Reciprocal  Precipitation.  A  further  deduction  of  the  elec- 
trical theory  of  precipitation  is,  that  when  two  oppositely-charged 
colloids  are  mixed,  they  should  precipitate  each  other,  and  the 
resulting  precipitate  should  contain  both  colloids.  Experiments 
carried  out  by  Biltz*  have  confirmed  these  predictions.  He 
showed  that  when  a  solution  of  a  positively-charged  colloid  is 
added  to  a  solution  of  a  negatively-charged  colloid,  precipitation 
occurs,  unless  the  quantity  of  the  added  colloid  is  either  relatively 
very  large  or  very  small.  He  also  showed  that  when  two  colloids 
of  the  same  electrical  sign  are  mixed  no  precipitation  occurs.  Just 
as  the  amount  of  precipitation  caused  by  the  addition  of  an  elec- 
trolyte to  a  sol  is  conditioned  by  the  rate  at  which  the  electrolyte 
is  added,  so  also  the  precipitation  of  one  colloid  by  another  is  de- 
*  Berichte,  37,  1095  (1904). 


COLLOIDS 


261 


pendent  upon  the  manner  in  which  the  two  sols  are  mixed.  The 
extent  to  which  one  sol  is  precipitated  by  another  sol  of  opposite 
sign  is  largely  determined  by  the  amount  of  one  that  is  added  to  a 
definite  amount  of  the  other.  This  is  clearly  shown  by  the  data 
of  the  following  table  which  give  the  results  obtained  by  Biltz  on 
adding  ferric  hydroxide  sol  to  2  cc.  of  an  antimony  trisulphide  sol 
containing  2.8  mg.  per  cc. 

PRECIPITATION  OF  COLLOIDAL  ANTIMONY  TRISULPHIDE 
BY   COLLOIDAL  FERRIC  HYDROXIDE. 


Fe2O3(mg.). 

Immediate  Result. 

Result  after  One  Hour. 

0.8 

Cloudy 

Almost  homogeneous 

3.2 

Small  flakes 

Unchanged 

4.8 

Flakes 

Yellow  liquid 

6.4 
8.0 

12.8 

Complete  precipitation 
Slow  precipitation 
Cloudy 

Complete  precipitation 
Complete  precipitation 
Slight  precipitation 

20.8 

Cloudy 

Homogeneous 

It  will  be  seen  that  the  addition  of  a  small  amount  of  ferric 
hydroxide  produces  hardly  any  precipitation.  With  the  addition 
of  larger  amounts  of  ferric  hydroxide,  the  amount  of  precipitation 
increases  until  finally  complete  precipitation  is  attained.  The 
addition  of  larger  quantities  of  ferric  hydroxide  produces  either 
little  or  no  precipitation.  It  has  been  found  that  at  the  concen- 
tration which  just  produces  complete  precipitation,  the  electrical 
charges  on  the  two  sols  are  equivalent.  When  the  amount  of  ferric 
hydroxide  exceeds  that  required  for  complete  precipitation,  it  is 
more  than  probable  that  the  particles  of  colloidal  antimony  tri- 
sulphide are  completely  enveloped  by  the  particles  of  ferric  hy- 
droxide and  thereby  rendered  inactive. 

When  we  come  to  study  the  action  of  one  emulsoid  on  another, 
we  find,  as  might  be  expected  from  the  general  behavior  of  emul- 
soids  toward  electrolytes,  that  the  phenomena  are  more  complex 
and  very  much  less  well-defined  than  with  suspensoids.  Although 
mutual  precipitation  does  take  place  with  emulsoids,  the  close 
resemblance  between  emulsoids  and  true  solutions  renders  the 
phenomenon  more  or  less  indistinct. 


262  THEORETICAL  CHEMISTRY 

When  a  suspensoid  sol  is  added  to  an  emulsoid  sol  having  an 
opposite  electric  charge,  precipitation  may  or  may  not  occur 
according  to  the  relative  amounts  of  the  two  colloids  in  the  mix- 
ture. When  the  two  colloids  are  present  in  electrically  equivalent 
quantities  precipitation  occurs,  otherwise  one  colloid  exerts  a 
protective  action  on  the  other. 

Characteristics  of  Gels.  Gels  are  generally  obtained  by  cooling 
or  evaporating  emulsoid  sols  and,  since  the  latter  are  known  to  be 
two-phase  liquid  systems,  it  is  natural  to  infer  that  gels  may  also 
be  two-phase  systems.  According  to  this  conception,  the  only 
difference  between  an  emulsoid  sol  and  a  gel  is,  that  in  the  latter 
the  concentration  of  at  least  one  of  the  phases  is  greatly  increased 
and  thereby  imparts  greater  viscosity  and  rigidity  to  the  system. 
It  has  been  suggested  that  the  more  concentrated  of  the  two  phases 
forms  the  walls  of  an  assemblage  of  cells  within  which  the  more 
dilute  phase  is  enclosed.  The  view  that  gels  possess  a  distinct 
cellular  structure  is  fully  confirmed  by  microscopic  examination. 

The  extreme  sensitiveness  of  gels  to  changes  in  temperature  and 
to  the  presence  of  extraneous  substances  renders  their  investi- 
gation exceedingly  difficult.  Notwithstanding  the  experimental 
difficulties  involved  in  the  study  of  gels,  sufficient  knowledge  has 
been  gained  of  their  properties  to  make  a  brief  account  of  these 
necessary  in  any  treatment  of  the  subject  of  colloids. 

Physical  Properties  of  Gels.  The  process  of  gel  formation 
from  a  dry  gelatinous  colloid  and  water  invariably  involves  con- 
traction. This  statement  should  not  be  confused  with  the  fact 
that  a  gel  on  immersion  in  water  undergoes  appreciable  increase  in 
volume. 

Gels  have  been  shown  by  Barus  *  to  be  considerably  more  com- 
pressible than  solids.  The  compressibility  increases  as  the  tem- 
perature is  raised  until,  when  the  gel  is  transformed  into  the  sol, 
the  compressibility  becomes  equal  to  that  of  pure  water.  The 
temperature  of  some  gels,  such  as  rubber  and  gelatine,  is  lowered 
by  compression  and  raised  by  tension. 

The  thermal  expansion  of  gels  is  nearly  identical  with  that  of  the 
more  fluid  component  of  the  gel. 

*  Am.  Jour.  Sci.,  6,  285  (1898). 


COLLOIDS  263 

The  rate  at  which  pure  substances  diffuse  in  gels  differs  only 
slightly  from  the  rate  of  diffusion  in  pure  water,  provided  the  con- 
centration of  the  gel  is  not  too  great:  The  slight  resistance  offered 
by  gels  to  diffusion  of  dissolved  substances  may  be  regarded  as 
further  evidence  in  favor  of  their  cellular  structure. 

The  modulus  of  elasticity  of  a  gel,  cast  in  a  cylindrical  mold,  is 
given  by  the  formula 

PI 


where  P  is  the  tension  which  produces  the  increase  in  length  AZ  in 
a  cylinder  whose  length  is  I  and  whose  radius  is  r.  It  has  been 
found  that  the  modulus  of  elasticity  in  gelatine  gels  increases  as  the 
square  of  the  concentration  of  the  gel.  The  time  of  recovery,  after 
releasing  the  tension,  increases  as  the  concentration  of  the  gel 
increases. 

The  shearing  modulus  for  a  gel  is  given  by  the  formula 


2(1+;*) 

where  /*  denotes  the  ratio  of  the  relative  contraction  of  the  diameter 
to  the  relative  change  in  length.  The  viscosity  of  a  gel  may  be 
calculated  from  the  shearing  modulus  by  means  of  the  equation 

•n  =  Esr, 

where  r  denotes  the  time  of  recovery.  Since  both  Es  and  r  increase 
with  the  concentration  of  the  gel  it  is  apparent  that  the  viscosity 
of  the  gel  must  also  increase  with  the  concentration. 

As  is  well  known,  when  glass  is  subjected  to  pressure  or  is  un- 
equally strained,  it  exhibits  the  phenomenon  of  double  refraction. 
Since  glass  bears  some  resemblance  to  gels  in  being  a  highly  viscous, 
supercooled  liquid,  it  might  reasonably  be  inferred  that  gels  should 
also  show  double  refraction.  Experiments  with  collodion  and 
gelatine  have  shown  that  these  substances,  when  subjected  to 
pressure,  behave  similarly  to  glass. 

Hydration  and  Dehydration  of  Gels.  The  complementary 
processes  of  hydration  and  dehydration  of  gels  are  extremely  inter- 


264  THEORETICAL  CHEMISTRY 

esting.  Following  Freundlich  *  we  will  consider  the  subject  very 
briefly  under  the  two  following  heads:  (a)  Non-elastic  Gels  and 
(b)  Elastic  Gels. 

(a)  Nvn-elastic  Gels.  When  freshly  prepared  aluminium  or  fer- 
ric hydroxide  gels  were  placed  in  a  desiccator,  it  was  found  by 
van  Bemmelen  f  that  the  rate  at  which  these  substances  lost  water 
was  continuous.  Furthermore,  on  removing  the  dried  gels  from 
the  desiccator  it  was  found  that  the  process  of  recovery  of  moisture 
was  also  continuous.  It  will  be  shown  in  a  subsequent  chapter 
(p.  320)  that  a  definite  hydrate  in  the  presence  of  its  products  of 
dissociation,  possesses  a  constant  vapor  pressure  so  long  as  any  of 
that  particular  hydrate  is  present.  The  fact  that  the  vapor 
pressure  of  the  gels  investigated  by  van  Bemmelen  did  not  remain 
constant  but  decreased  continuously  as  the  water  was  removed, 
proved  conclusively  that  no  chemical  compounds  were  formed  in 
the  process  of  gel  hydration. 

Another  gel  which  has  been  made  the  subject  of  much  careful 
investigation  is  silicic  acid.  The  dehydration  curve  of  silicic  acid 
is  continuous,  with  the  exception  of  a  short  portion  where  an  ap- 
preciable amount  of  water  is  lost  without  much  of  any  change  in 
the  vapor  pressure.  This  portion  of  the  curve  corresponds  to  a 
marked  change  in  the  appearance  of  the  gel.  The  gel  which  had 
hitherto  been  clear  and  transparent  became  opalescent  'soon  after 
the  vapor  pressure  had  attained  a  temporarily  constant  value. 
The  opalescence  gradually  permeated  the  entire  mass,  until  the 
gel  acquired  a  yellow  color  by  transmitted  light,  and  a  bluish  color 
by  reflected  light.  These  colors  suggest  a  marked  increase  in  the 
degree  of  dispersity  of  the  gel,  an  inference  the  correctness  of 
which  subsequent  investigation  has  fully  confirmed. 

The  curve  of  hydration,  while  resembling  the  curve  of  dehydra- 
tion in  many  respects,  departs  quite  widely  from  it  in  others.  The 
change  in  dispersity,  as  indicated  by  the  appearance  in  reverse 
order  of  the  color  and  opalescent  phenomena  mentioned  above, 
also  was  manifest. 

*  Kapillarchemie,  p.  486. 

t  Zeit.  anorg.  Chemie,  5,  466  (1894);  13,  233  (1897);  18,  14,  98  (1898);  30, 
265  (1902). 


COLLOIDS  265 

(b)  Elastic  Gels.  The  absence  in  elastic  gels  of  a  horizontal 
portion  in  the  dehydration  curves,  and  the  fact  that  although 
elastic  gels  may  have  become  saturated  with  water  vapor,  they 
still  retain  the  power  of  taking  up  large  amounts  of  liquid  water 
when  immersed  in  that  medium,  constitute  the  chief  differences 
between  elastic  and  non-elastic  gels. 

The  amount  of  water  which  can  be  taken  up  by  an  elastic  gel  is 
exceedingly  large.  Thus,  on  exposing  a  plate  of  gelatine  weighing 
0.904  gram  for  eight  days  in  an  atmosphere  saturated  with  water 
vapor,  Schroeder  *  found  that  it  had  taken  up  0.37  gram  of  water. 
On  exposing  for  a  longer  period  of  time  under  the  same  conditions, 
he  found  no  further  gain  in  weight,  and  on  removing  the  gelatine 
plate  from  the  moist  atmosphere  and  placing  it  in  a  desiccator,  the 
plate  slowly  gave  up  the  absorbed  moisture  and  regained  its  origi- 
nal weight.  On  the  other  hand,  when  the  plate,  after  having 
absorbed  the  maximum  weight  of  moisture  from  the  air,  was  im- 
mersed in  water,  it  was  found  to  increase  in  weight  very  rap- 
idly. Thus,  on  immersing  the  above  plate  which  weighed  1.274 
grams  when  saturated  in  moist  air,  and  allowing  it  to  remain 
in  water  for  one  hour,  it  was  found  to  have  taken  up  5.63 
grams  of  water.  After  an  immersion  of  twenty-four  hours,  the 
plate  was  found  to  have  taken  up  the  maximum  weight  of  water 
it  was  capable  of  absorbing  at  that  temperature.  On  remov- 
ing the  plate,  it  was  found  to  part  with  the  absorbed  water 
very  readily,  even  in  moist  air,  the  greater  part  of  the  absorbed 
water  being  so  loosely  held  that  the  vapor  pressure  of  the  gel 
remained  the  same  as  that  of  pure  water  at  the  temperature  of 
the  experiment. 

It  is  impossible  to  measure  directly  the  pressure  produced  by  gels 
when  they  take  up  water,  but  some  idea  of  the  magnitude  of  these 
pressures  may  be  obtained  by  coating  a  glass  plate  with  gelatine, 
which  has  imbibed  the  maximum  amount  of  water,  and  observing 
the  degree  to  which  the  glass  plate  is  bent  by  the  drying  gelatine 
film.  Frequently  the  elastic  limit  of  the  glass  is  exceeded  and 
the  plate  breaks  under  the  stress  produced  by  the  dehydration  of 
the  gel. 

*  Zeit.  phys.  Chem.,  45,  75  (1903). 


266 


THEORETICAL  CHEMISTRY 


Velocity  of  Imbibition.  The  rate  at  which  gels  imbibe  water 
has  been  studied  by  Hofmeister.*  The  velocity  of  imbibition  of 
water  by  thin  plates  of  gelatine  and  agar-agar  was  measured  by 
removing  the  plates  at  definite  intervals  and  determining  their 
increase  in  weight.  Owing  to  the  time  consumed  in  making  the 
weighings,  the  time  intervals  are  affected  by  an  appreciable  error. 
The  following  table  gives  the  data  of  a  single  experiment  with 
gelatine. 

VELOCITY  OF  IMBIBITION   IN  GELATINE. 
(Thickness  of  plate  0.5  mm.) 


Time  (min.). 

Water  Imbibed 
(grams). 

I 

5 

3.08 

0.090 

10 

3.88 

0.084 

15 

4.26 

0.084 

20 

4.58 

0.064 

25 

4.67 

0.075 

00 

4.96 



If  the  weight  of  water  imbibed  in  t  minutes  is  wt,  and  w^  is  the 
maximum  weight  of  water  which  a  gel  can  take  up  under  the  con- 
ditions, then  the  velocity  of  imbibition  should  be  given  by  the 
equation 

dw         , 

-~jj  =  k(wM-  wt), 

which  on  integration  becomes 


t        (wx  -  wt) 

The  figures  in  the  third  column  of  the  preceding  table  were  cal- 
culated by  means  of  this  equation  and,  although  the  values  are  not 
strictly  constant,  the  variation  is  no  greater  than  might  be  expected 
where  the  experimental  error  is  so  large. 

Heat  of  Imbibition.  The  process  of  imbibition  is  accom- 
panied by  an  evolution  of  heat.  Quantitative  measurements 

*  Arch.  exp.  Pathol.  u.  Pharmakol.,  27,  395  (1890). 


COLLOIDS 


267 


of  the  heat  evolved  when  gels  take  up  water  have  been  made 
by  Wiedemann  and  Ludeking  *  and  also  by  Rodewald.f  The 
following  table  gives  Rodewald's  data  on  the  heat  of  imbibition 
of  starch. 

HEAT  OF  IMBIBITION  OF  STARCH. 

(Weight  of  dry  starch  100  grams.) 


Per  Cent,  Water. 

Heat  in  Calories  per 
Gram  of  Starch. 

0.23 

28.11 

2.39 

22.60 

6.27 

15.17 

11.65 

8.43 

15.68 

5.21 

19.52 

2.91 

It  will  be  observed  that  the  greatest  development  of  heat  ac- 
companies the  initial  stages  of  imbibition  where  very  small 
amounts  of  water  are  taken  up.  This  is  what  we  might  expect 
when  we  remember  that  it  is  the  last  remaining  portion  of  water 
which  is  most  difficult  to  remove  from  a  gel,  and  that  it  is  only 
through  the  application  of  heat  that  its  complete  removal  can  be 
effected. 

Imbibition  in  Solutions.  When  a  gel  is  immersed  in  a  saline 
solution,  the  salt  distributes  itself  between  the  solvent  and  the 
gel.  The  rate  of  imbibition  is  found  to  vary  greatly  according 
to  the  salt  which  is  present  in  the  solution.  Thus,  the  velocity 
of  imbibition  has  been  found  to  be  accelerated  by  the  presence 
of  the  chlorides  of  ammonium,  sodium  and  potassium  and  by 
the  nitrate  and  bromide  of  sodium ;  on  the  other  hand,  the  pres- 
ence of  the  nitrate,  sulphate  and  tartrate  of  sodium  retard  im- 
bibition. 

The  effect  of  acids  and  bases  on  imbibition  appears  to  be  similar 
to  the  influence  of  salts. 

Adsorption.  The  change  in  concentration  which  occurs  at 
the  boundary  between  two  heterogeneous  phases  is  termed  adsorp- 

*  Wied.  Ann.,  25,  145  (1885). 

t  Zeit.  phys.  Chem.,  24,  206  (1897). 


268  THEORETICAL  CHEMISTRY 

tim.  At  the  surface  of  a  solid  surrounded  by  a  gas  or  vapor,  the 
phenomenon  is  generally  known  as  gaseous  adsorption,  since  any 
difference  which  may  occur  in  the  concentration  of  the  solid  phase 
is  much  too  small  to  be  detected.  At  the  boundary  between 
liquid  and  gaseous  phases,  the  concentration  of  each  phase  un- 
doubtedly undergoes  alteration.  In  the  case  of  the  boundary,  be- 
tween solid  and  liquid  phases,  the  only  apparent  inequality  in 
concentration  occurs  on  the  liquid  side  of  the  boundary,  notwith- 
standing the  fact  that  the  adsorbed  substance  is  quite  commonly 
regarded  as  being  bound  to  the  surface  of  the  solid  phase.  The 
cause  of  this  erroneous  conception  is,  that  the  extremely  thin  layer 
of  liquid  in  which  the  alteration  in  concentration  actually  occurs, 
is  the  layer  which  wets  the  surface  of  the  solid  and  hence  is  the 
layer  which  adheres  to  the  solid  when  it  is  removed  from  the 
liquid. 

The  retention  of  gases  by  charcoal  is  a  typical  example  of  gaseous 
adsorption,  while  the  removal  of  coloring  matter  by  charcoal  in  the 
purification  of  various  organic  substances  may  be  cited  as  an 
example  of  adsorption  of  a  liquid  by  a  solid. 

If  the  adsorbed  substance  increases  in  concentration  in  the 
vicinity  of  the  boundary,  the  adsorption  is  said  to  be  positive;  if  it 
decreases,  the  adsorption  is  said  to  be  negative. 

Adsorption  of  Gases.  In  gases,  adsorption-equilibrium  is 
attained  with  remarkable  rapidity.  Thus,  if  a  gas  is  admitted 
into  a  vessel  containing  some  freshly  prepared  cocoanut  charcoal, 
the  pressure  will  fall  immediately  to  a  value  which  corresponds  to 
the  removal  of  the  entire  adsorbed  volume  of  gas. 

The  concentration  of  adsorbed  gas  on  the  surface  of  a  solid,  when 
equilibrium  is  attained,  has  been  shown  to  be  approximately 
1  X  10~7  gram  per  square  centimeter.  This  value  is  of  the  same 
order  of  magnitude  as  the  strength  of  the  limiting  capillary  layer 
of  a  liquid  and,  therefore,  lends  support  to  the  suggestion  put  for- 
ward some  years  ago  by  Faraday  that  an  adsorbed  film  of  gas  may 
be  present  in  the  liquid  state. 

The  amount  of  gas  adsorbed  by  a  solid  increases  with  the  pres- 
sure and  diminishes  with  increasing  temperature.  The  following 
empirical  equation,  expressing  the  relation  between  the  amount  of 


COLLOIDS  269 

gas  adsorbed  by  a  solid  and  the  pressure,  has  been  proposed  by 
Freundlich  * 

x  -1 

—  =  apn , 
m 

where  x  is  the  total  mass  of  gas  adsorbed  on  a  surface  of  m  sq. 
cm.  under  a  pressure  p,  and  where  a  and  n  are  constants.  This 
equation,  known  as  the  adsorption  isotherm,  has  been  found  to  hold 
quite  generally. 

Adsorption  in  Solutions.  The  adsorption  phenomena  occur- 
ring at  the  surface  of  contact  of  a  solid  with  a  solution  are  similar 
to  the  phenomena  which  have  just  been  discussed.  Because  of  the 
frequency  of  its  occurrence  in  many  of  the  more  common  operations 
of  both  laboratory  and  factory,  the  subject  of  adsorption  in  solu- 
tions deserves  fuller  treatment. 

The  general  characteristics  of  adsorption  in  solutions  may  be 
briefly  summarized  as  follows :  — 

(1)  Adsorption  in  solutions  is  generally  positive,  i.e.,  on  shaking 
a  solution  with  a  finely-divided  adsorbent,  the  volume  concentra- 
tion of  the  solution  will  diminish. 

(2)  The  amount  of  positive  adsorption  may  be  sufficient  to  re- 
move almost  all  of  the  solute  from  a  solution,  especially  if  the  solu- 
tion is  dilute.     On  the  other  hand,  negative  adsorption  is  always 
very  small,  and  frequently  is  immeasurable. 

(3)  Adsorption  is  directly  proportional  to  the  so-called  "  specific 
surface,"  the  latter  term  being  defined  as  the  ratio  of  the  total. sur- 
face of  the  adsorbent  to  its  volume. 

(4)  On  shaking  a  definite  weight  of  an  adsorbent  with  a  given 
volume  of  solution  of  known  concentration,  a  definite  equilibrium 
will  be  established.     If  the  solution  is  then  diluted  with  a  known 
amount  of  solvent,  the  adsorption  will  decrease  until  it  acquires  the 
same  value  which  it  would  have  attained,  had  the  same  weight  of 
adsorbent  been  introduced  directly  into  the  more  dilute  solution. 
For  example,  if  1  gram  of  charcoal  is  agitated  with  100  cc.  of  a 
0.0688  molar  solution  of  acetic  acid  for  20  hours,  adsorption  is 
found  to  reduce  the  original  concentration  of  the  acid  to  0.0678 
molar. 

*  Zeit.  phys.  Chem.,  57,  385  (1906). 


270  THEORETICAL  CHEMISTRY 

In  a  second  experiment,  if  1  gram  of  charcoal  is  shaken  for  the 
same  period  of  time  with  50  cc.  of  2  X  0.0688  =  0.1376  molar 
acetic  acid,  and  then,  after  adding  50  cc.  of  water,  the  shaking  is 
continued  for  an  additional  period  of  3  hours,  the  final  concentration 
of  the  acid  will  be  found  to  be  the  same  as  in  the  first  experiment. 

(5)  It  is  impossible  to  determine  the  specific  surface  of  an  ad- 
sorbent directly  owing  to  its  porosity.     However,  according  to  (3) 
adsorption  is  directly  proportional  to  the  specific  surface  and  there- 
fore the  weights  of  different  adsorbents  which  produce  the  same 
amount  of  adsorption  may  be  assumed  to  possess  equal  specific 
surfaces. 

(6)  Adsorption  in  solution  is  largely  dependent  upon  the  surface 
tension  of  the  solvent.     In  solutions  of  the  same  substance  in 
different  solvents,  the  greatest  adsorption  occurs  in  that  solution 
whose  solvent  possesses  the  highest  surface  tension. 

(7)  The  order  of  efficiency  of  adsorption  is  not  only  independent 
of  the  nature  of  the  solvent  but  also  of  the  nature  of  the  adsorbed 
substance. 

The  Adsorption  Isotherm.  The  empirical  equation  of  Freund- 
lich  for  gaseous  adsorption  has  been  found  to  apply  equally  well  to 
adsorption  equilibria  in  solutions.  The  equation  may  be  written 
as  follows : 

x          1 

—  =  acn , 
ra 

where  x  is  the  weight  of  substance  adsorbed  by  a  weight  m  of  ad- 
sorbent from  a  solution  whose  volume-concentration  at  equilibrium 
is  c,  and  where,  as  before,  a  and  n  are  constants.  The  constant  n 
varies  in  different  cases  from  n  =  2  to  n  =  10;  within  these  limits 
the  value  of  n  is  independent  of  the  temperature  and  also  of  the 
natures  of  the  adsorbed  substances  and  the  adsorbent.  Although 
the  value  of  the  constant  a  varies  over  a  wide  range,  the  ratio  of  its 
values  for  two  adsorbents  in  different  solutions  is  practically  con- 
stant. 

The  following  table  contains  the  data  given  by  Freundlich  *  on 
the  adsorption  of  acetic  acid  by  charcoal. 

*  Kapillarchemie,  p.  147. 


COLLOIDS 


271 


ADSORPTION  OF  AQUEOUS  ACETIC  ACID  BY  CHARCOAL. 
t  =  25°;        a  =  2.606;        l/n  =  0.425. 


Concentration 
(mols  per  liter). 

x/m  (obs.). 

x/m  (calc.). 

0.0181 

0.467 

0.474 

0.0309 

0.624 

0.596 

0.0616 

0.801 

0.798 

0.1259 

1.11 

1.08 

0.2677 

1.55 

1.49 

0.4711 

2.04 

1.89 

0.8817 

2.48 

2.47 

2.785 

3.76 

4.01 

The  validity  of  the  adsorption  isotherm  is  best  tested  graphically, 
by  plotting  the  logarithms  of  the  experimentally  determined  values 
of  x/m  against  the  logarithms  of  the  corresponding  concentrations. 
If  the  equation  holds,  a  straight  line  should  be  obtained.  The 
curves  shown  in  Fig.  69  represent  x/m  as  a  function  of  c,  and 
log  x/m  as  a  function  of  log  c :  it  will  be  observed  that  the  loga- 
rithmic plot  is  practically  rectilinear. 

log  C 


c 
Fig.  69. 

Surface  Energy  of  Colloids.  In  almost  all  colloidal  solutions 
there  exists  a  difference  of  potential  between  the  particles  of  the  col- 
loid and  the  surrounding  medium.  The  importance  of  this  factor 


272  THEORETICAL  CHEMISTRY 

in  interpreting  the  behavior  of  colloids  has  already  been  empha- 
sized. Another  factor  of  equal  importance  in  connection  with 
colloidal  phenomena,  is  that  which  depends  upon  the  enormous 
surface  of  contact  between  the  colloid  and  the  surrounding  medium. 
There  is  an  abundance  of  evidence  showing  that  a  colloidal  solution 
is  non-homogeneous,  or  in  other  words,  that  it  is  essentially  a  sus- 
pension of  finely-divided  particles  in  a  fluid  medium.  An  immense 
increase  in  superficial  area  results  from  the  division  and  sub- 
division of  matter.  To  bring  about  this  comminution  requires  a 
large  expenditure  of  energy.  In  a  colloidal  solution  this  energy  is 
stored  up  in  the  colloidal  particles  in  the  form  of  surface  energy, 
which  may  be  defined  as  the  product  of  surface  area  and  surface 
tension. 

For  example,  suppose  1  cc.  of  a  substance  to  be  reduced  to 
cubical  particles  measuring  0.1  /i  on  each  edge,  and  let  the  particles 
be  suspended  in  water  at  17°  C.  The  total  energy  involved  can 
be  calculated  as  follows:  —  The  volume  of  a  single  particle  is 
0.1  ij?  or  [1  X  10~15  cc. ;  hence  the  total  number  of  particles  is 
1  X  1015.  The  surface  of  a  single  particle  is  6  X  (0.1  ju)2,  or 
6  X  10~10  sq.  cm.,  and  the  total  surface  is  6  X  105  sq.  cm.  The 
surface  tension  of  water  at  17°  C.  is  71  dynes;  hence  the  total 
surface  energy  is  71  X  6  X  105  =  4.32  X  107  ergs.  This  enormous 
figure  shows  that  where  the  surface  of  the  disperse  phase  is  highly 
developed,  as  it  is  in  colloidal  solutions,  the  surface  energy  becomes 
a  very  important  factor  in  determining  the  behavior  of  the  system. 
This  is  especially  the  case  when  the  degree  of  aggregation  of  the 
colloidal  particles  is  changed,  since  a  relatively  small  change  in  the 
amount  of  aggregation  may  involve  a  great  change  in  the  surface 
exposed  and  a  corresponding  change  in  the  surface  energy.  A  very 
close  connection  exists  between  the  electrical  and  surface  factors  in 
a  colloidal  solution. 

Surface  Concentration.  It  has  been  pointed  out  in  an  earlier 
chapter  (p.  144)  that  as  the  result  of  unbalanced  molecular  attrac- 
tion, the  surface  of  a  liquid  behaves  like  a  tightly  stretched  mem- 
brane. In  consequence  of  this  contractile  force,  or  surface  tension, 
the  pressure  at  the  surface  of  a  liquid  is  greater  than  the  pressure 
within  the  liquid. 


COLLOIDS  273 

The  experiments  of  Soret  (p.  208)  and  the  theoretical  deductions 
of  van't  Hoff  have  shown  that  when  a  dilute  solution  is  unequally 
heated,  the  solute  distributes  itself  in  accordance  with  the  gas  laws, 
the  solution  becoming  more  concentrated  in  the  cooler  portion. 
Just  as  the  homogeneity  of  a  dilute  solution  has  been  shown  to 
be  disturbed  by  inequality  of  temperature,  so  also  inequality  of 
pressure  may  be  assumed  to  cause  differences  in  concentration  in 
the  solution.  Although  direct  experimental  verification  is  difficult, 
there  is  abundant  evidence  for  the  view  that  the  concentration  at 
the  surface  of  solution  differs  from  the  volume-concentration  of 
the  solution  in  consequence  of  the  greater  pressure  in  the  surface 
layer. 

The  mathematical  relation  between  surface  concentration  and 
surface  tension  was  first  deduced  by  J.  Willard  Gibbs  *  in  1876. 
The  following  simplified  derivation  of  this  important  equation  is 
due  to  Ostwald.  Let  s  be  the  surface  of  a  solution  whose  surface 
tension  is  7,  and  let  it  be  assumed  that  the  surface  contains  1  mol 
of  the  solute.  If  a  very  small  portion  of  the  solute  enters  the  sur- 
face layer  from  the  solution,  thereby  causing  a  diminution  dy  in 
the  surface  tension,  the  corresponding  change  in  energy  will  be 
s  dy.  But  this  gain  in  energy  must  be  equivalent  to  the  osmotic 
work  involved  in  effecting  the  removal  of  the  same  weight  of  solute 
from  the  solution.  Let  v  be  the  volume  of  solution  containing 
unit  weight  of  solute,  and  let  dp  be  the  difference  in  the  osmotic 
pressures  of  the  solution  before  and  after  its  removal:  the  osmotic 
work  will  be  —  v  dp.  Since  the  gain  in  surface  energy  and  the 
osmotic  work  are  equal,  we  have 

s  dy  =  —v  dp. 

The  solutions  being  dilute,  we  may  assume  that  the  gas  laws  hold, 
and  since  v  =  RT/p,  we  may  write 

RT, 

sdy  =  -   — dp, 

dy  RT 

or  -• 

dp  sp 


Trans.  Conn.  Acad.,  Vol.  Ill,  439  (1876). 


274  THEORETICAL  CHEMISTRY 

Since  pressure  is  directly  proportional  to  concentration,  the  pre- 
ceding equation  becomes 

dry  =  _RT 
dc  sc 

But  s  has  already  been  defined  as  the  surface  which  contains  1  mol 
of  solute  in  excess,  from  which  it  follows  that  the  excess  of  solute 
in  unit  surface  is  1/s.  Writing  u  =  1/s,  we  have 

c    dy 
-RTW 
which  is  the  equation  of  Gibbs. 

From  this  equation  it  is  evident  that  if  the  surface  tension,  7, 
increases  with  the  concentration,  then  u  is  negative  and  the  surface 
concentration  is  less  than  the  concentration  of  the  bulk  of  the 
solution.  This  is  clearly  negative  adsorption.  On  tha  other 
hand,  if  7  decreases  as  the  concentration  increases,  u  is  positive  and 
the  surface  concentration  is  greater  than  the  concentration  of  the 
bulk  of  the  solution,  or  the  adsorption  is  positive.  Finally,  if  the 
surface  tension  is  independent  of  the  concentration,  then  the  con- 
centration of  the  solute  in  both  the  surface  layer  and  the  bulk  of 
the  solution  will  be  the  same. 

Preparation  of  Colloidal  Solutions.  Since  1861,  when  Graham 
published  his  first  paper  on  colloids,  numerous  investigators  have 
devised  methods  for  the  preparation  of  colloidal  solutions.  Within 
recent  years  our  knowledge  of  this  class  of  solutions  has  been 
greatly  increased,  many  crystalloidal  substances  having  been 
obtained  in  the  colloidal  condition.  As  a  result  of  these  investi- 
gations, we  no  longer  speak  of  crystalloidal  and  colloidal  matter, 
but  use  the  terms  crystalloid  and  colloid  to  distinguish  two  dif- 
ferent states.  In  fact  it  is  now  recognized  that  it  is  simply  a 
matter  of  overcoming  certain  experimental  difficulties,  before  it 
will  be  possible  to  obtain  all  forms  of  matter  in  the  colloidal 
state.  The  scope  of  this  book  forbids  a  detailed  account  of  the 
various  methods  which  have  been  devised  for  the  preparation  of 
colloidal  solutions.*  We  must  content  ourselves  with  a  general 
classification  of  these  methods  into  two  groups  as  follows :  — 

*  See  "Die  Methoden  zur  Herstellung  Kolloider  Losungen  anorganischer 
Stoffe,"  by  Theodore  Svedberg,  Dresden,  1909. 


COLLOIDS  275 

(1)  Crystallization  Methods,  and  (2)  Solution  Methods.  These 
two  divisions  are  sufficiently  comprehensive  to  include  all  of  the 
known  methods  for  the  preparation  of  colloidal  solutions,  with  the 
possible  exception  of  the  electrical  methods  which  may  be  con- 
sidered as  forming  a  separate  group. 

Crystallization  Methods.  The  crystallization  methods  include 
the  following  subdivisions :  - 

(1)  Methods  involving  cooling  of  a  liquid  or  solution. 
Example:  —  On  cooling  an  alcoholic  solution  of  sulphur  in  liquid 

air,  a  transparent,  highly  dispersed,  solid  sol  is  obtained. 

(2)  Methods  involving  change  of  medium. 

Example:  —  On  gradually  adding  a  solution  of  mastic  in  alcohol 
to  a  large  volume  of  water,  the  mastic  is  precipitated  in  a  finely 
divided  condition  and  a  colloidal  mastic  hydrosol  results. 

(3)  Reduction  methods. 

Example:  —  On  adding  a  cold,  dilute  solution  of  hydrazine 
hydrate  to  a  dilute,  neutral  solution  of  auric  chloride,  a  dark  blue 
gold  sol  is  obtained. 

In.  addition  to  hydrazine,  numerous  other  reducing  agents  may 
be  employed,  such  as  phosphorus,  carbon  monoxide,  hydrogen, 
acetylene,  formaldehyde,  acrolein,  various  carbohydrates,  hy- 
droxylamine,  phenylhydrazine,  and  metallic  ions. 

(4)  Oxidation  methods. 

Example:  —  On  oxidizing  a  solution  of  hydrogen  sulphide  by  air 
or  sulphur  dioxide,  a  colloidal  solution  of  sulphur  is  obtained. 

(5)  Hydrolysis  methods. 

Example:  —  When  a  solution  of  ferric  chloride  is  slowly  added  to 
a  large  volume  of  boiling  water,  the  salt  undergoes  hydrolysis,  and 
on  cooling  the  dilute  solution,  a  reddish-brown  ferric  hydroxide 
sol  is  obtained. 

(6)  Methods  involving  metathesis. 

Example:  —  A  colloidal  solution  of  silver  may  be  prepared  by 
adding  a  few  drops  of  a  dilute  solution  of  sodium  chloride  to  a  dilute 
solution  of  silver  nitrate,  provided  the  resulting  solution  of  sodium 
nitrate  is  below  the  precipitating  concentration. 


276  THEORETICAL  CHEMISTRY 

Solution  Methods.  Under  this  heading  are  to  be  grouped  the 
so-called  " peptization"  methods.  The  term,  peptization,  was  intro- 
duced bv  Graham  to  express  the  transformation  of  a  gel  into  a  sol. 
To-day  we  understand  a  peptizer  to  be  a  substance  which,  if 
sufficiently  concentrated,  is  capable  of  effecting  the  solution  of  a 
solid  which  is  insoluble  in  its  dispersion  medium.  A  typical 
example  of  peptization  is  afforded  by  silver  chloride  which  forms  a 
sol  on  prolonged  digestion  with  a  solution  which  contains  either 
Ag*  or  Cl'.  It  is  apparent  that  the  rate  of  peptization  can  be  con- 
trolled by  dilution  of  the  peptizer,  and  that  when  the  sol  stage  is 
attained,  the  peptizer  may  be  readily  removed  by  dialysis. 

Numerous  reactions  are  known  in  which  the  conversion  of  an 
insoluble  precipitate  into  a  sol  can  only  be  effected  through  the 
removal  of  the  excess  of  electrolyte  by  prolonged  washing  or  dial- 
ysis. A  familiar  example  of  this  type  of  peptization  is  furnished 
by  the  tendency  of  many  precipitates  to  run  through  the  filter 
after  too  prolonged  washing  with  water. 

Electrical  Methods.  These  methods  of  preparing  colloidal  solu- 
tions depend  upon  the  dispersive  action  of  a  powerful  electric 
discharge  upon  compact  metals.  In  1897  Bredig  *  discovered, 
while  studying  the  action  of  the  electric  current  on  different 
liquids,  that  if  an  arc  be  established  between  two  metallic  wires 
immersed  in  a  liquid,  minute  particles  of  metal  are  torn  off  from 
the  negative  terminal  and  remain  suspended  in  the  liquid  indefi- 
nitely. In  order  to  prepare  a  colloidal  solution  by  the  method  of 
electrical  dispersion,  Bredig  recommends  that  a  direct  current 
arc  be  established  between  wires  of  the  metal  of  which  a  colloidal 
solution  is  desired,  the  ends  of  the  wires  being  submerged  in  water 
in  a  well-cooled  vessel,  as  shown  in  Fig.  70.  The  current  employed 
ranges  in  strength  from  5  to  10  amperes,  and  the  voltage  lies  be- 
tween 30  and  110  volts.  A  rheostat  and  an  ammeter  are  included 
in  the  circuit. 

The  wires  are  brought  in  contact  for  an  instant  in  order  to 

establish  the  arc,  after  which  they  are  separated  about  2  mm. 

During  the  gentle  hissing  of  the  arc,  clouds  of  colloidal  metal  are 

projected  out  into  the  water  from  the  negative  wire,  a  portion  of 

*  Zeit.  Elektrochem.,  4,  514  (1897);  Zeit.  phys.  Chem.,  31,  258  (1899). 


COLLOIDS 


277 


the  metal  torn  off  being  distributed  through  the  water  as  a  coarse 
suspension.  The  size  of  the  particles  disrupted  from  the  negative 
terminal  is  dependent  upon  the  strength  of  the  current,  a  current 


Ammeter 


Fig.  70. 

of  10  amperes  producing  a  greater  proportion  of  colloidal  metal 
than  a  current  of  5  amperes.  The  addition  of  a  trace  of  potassium 
hydroxide  to  the  water  has  been  shown  to  facilitate  the  process  of 
dispersion.  When  gold  wires  are  used,  deep  red  colloidal  solu- 
tions are  obtained,  which  after  standing  for  several  weeks,  acquire 
a  bluish-violet  color.  With  extra  precautions,  the  red  colloidal 
gold  solutions  may  be  preserved  for  two  years.  These  solutions 
have  been  shown  by  Bredig  to  contain  about  14  mg.  of  gold  per 
100  cc.  In  this  manner  Bredig  prepared  colloidal  solutions  of 
platinum,  palladium,  iridium,  and  silver.  The  method  of  Bredig 
has  been  improved  and  extended  by  Svedberg. 

A  diagram  of  Svedberg's  apparatus  is  shown  in  Fig.  71.  The 
secondary  terminals  of  an  induction  coil,  capable  of  giving  a  spark 
ranging  from  12  to  15  cm.  in  length,  are  connected  in  parallel  with 
the  electrodes  and  a  glass  plate-condenser  having  a  surface  of 
approximately  225  sq.  cm.  Minute  fragments  or  grains  of  the 
metal  of  which  a  sol  is  desired  are  placed  on  the  bottom  of  the 
vessel  containing  the  dispersion  medium.  The  electrodes,  which 
need  not  necessarily  be  of  the  same  metal,  are  immersed  as  shown 
in  the  diagram,  and  during  the  process  of  electrical  dispersion,  the 
contents  of  the  vessel  are  gently  stirred  with  one  or  the  other  of  the 


278 


THEORETICAL  CHEMISTRY 


electrodes.  With  this  apparatus  Svedberg  has  succeeded  in  pre- 
paring colloidal  solutions  of  tin,  gold,  silver,  copper,  lead,  zinc, 
cadmium,  carbon,  silicon,  selenium,  and  tellurium.  He  has  also 


Induction  Coil 


Fig.  71. 

obtained  all  of  the  alkali  metals  in  the  colloidal  state,  ethyl  ether 
being  used  as  the  dispersion  medium.  An  interesting  observation 
made  by  Svedberg  in  the  course  of  his  experiments  is  that  the  color 
of  a  metal  is  the  same  in  both  the  colloidal  and  gaseous  states. 


CHAPTER  XIII. 
MOLECULAR  REALITY. 

The  Brownian  Movement.  If  a  liquid  in  which  fine  particles 
of  matter  are  suspended,  such  as  an  aqueous  suspension  of  gam- 
boge, be  examined  under  the  microscope,  the  suspended  particles 
will  be  seen  to  be  in  a  state  of  ceaseless,  erratic  motion.  This 
phenomenon  was  first  observed  in  1827  by  the  English  botanist, 
Robert  Brown,  while  examining  a  suspension  of  pollen  grains,  and 
has  been  called  the  Brownian  Movement  in  honor  of  its  discoverer. 

Ever  since  its  discovery,  the  Brownian  Movement  has  been  the 
subject  of  numerous  investigations.  It  was  not  until  1863,  how- 
ever, that  Wiener  suggested  that  the  cause  of  the  phenomenon  was 
the  actual  bombardment  of  the  suspended  particles  by  the  mole- 
cules of  the  suspending  medium.  Twenty-five  years  later  a  similar 
conclusion  was  reached  independently  by  Gouy,  who  showed  that 
neither  light  nor  convection  currents  within  the  liquid  could 
possibly  give  rise  to  the  motion.  Furthermore,  Gouy  showed  the 
movement  to  be  independent  of  external  vibration  and  only  slightly 
influenced  by  the  nature  of  the  suspended  particles.  The  smaller 
the  particles  and  the  less  viscous  the  suspending  medium,  the  more 
rapid  the  motion  was  found  to  be.  By  far  the  most  striking  feature 
of  the  phenomenon,  however,  is  the  fact  that  the  motion  is  cease- 


Perrin's  Experiments.  The  first  quantitative  investigation 
of  the  Brownian  Movement  was  undertaken  by  Perrin  in  1909. 
It  has  been  shown  (p.  100)  that  the  mean  kinetic  energy  Ek  of  one 
mol  of  a  perfect  gas  is  given  by  the  expression 

Ek  =  t  pv.  (1) 

Since  pv  =  RT,  we  may  write 

tf*  =  »7p  (2) 

279 


280  THEORETICAL  CHEMISTRY 

where  N  denotes  the  Avogadro  Constant,  i.e.,  the  number  of 
molecules  contained  in  one  mol  of  any  gas.  It  is  evident  that  if 
Ek  can  be  measured,  equation  (2)  affords  a  means  of  calculating  N, 
provided  we  are  warranted  in  applying  an  equation  which  has  been 
derived  for  the  gaseous  state  to  a  suspension  of  fine  particles  in  a 
liquid  medium.  It  has  already  been  shown  that  the  simple  gas 
laws  hold  for  dilute  solutions  and  therefore  we  may  assume  that, 
at  the  same  temperature,  the  mean  kinetic  energy  of  the  dissolved 
molecules  is  equal  to  that  of  the  gaseous  molecules.  In  other 
words,  at  the  same  temperature,  the  mean  kinetic  energy  of  all 
the  molecules  of  all  fluids  is  the  same,  and  is  directly  proportional 
to  the  absolute  temperature.  Since  the  gas  laws  apply  equally 
well  to  dilute  solutions  containing  either  large  or  small  molecules, 
Perrin  held  that  there  was  no  a  priori  reason  for  assuming  that  the 
grains  of  a  suspension  should  not  conform  to  the  same  laws.  If 
this  assumption  be  correct,  the  grains  of  a  uniform  suspension 
should  so  distribute  themselves  under  the  influence  of  gravity 
.  that,  when  equilibrium  is  attained,  the  lower  layers  will  have  a 
higher  concentration  than  the  upper  layers.  In  other  words,  the 
distribution  should  be  strictly  analogous  to  the  distribution  of  the 
air  over  the  surface  of  the  earth,  the  density  being  greatest  at  the 
surface  and  diminishing  as  the  altitude  increases. 

Let  us  imagine  a  suspension  to  be  confined  within  a  tall 
vertical  cylinder  whose  cross-sectional  area  is  s  sq.  cm.  As- 
suming that  the  suspension  has  come  to  equilibrium  under  the 
influence  of  gravitation,  let  n  be  the  number  of  grains  per  unit  of 
volume  at  a  height  h  from  the  base  of  the  cylinder.  Since  the 
concentration  diminishes  as  the  height  increases,  the  number  of 
grains  at  a  height  h  +  dh  will  be  n  —  dn.  The  osmotic  pressure 
of  the  grains  at  the  height  h  will  be  f  nEk,  where  Ek  is  the  mean 
kinetic  energy  of  each  grain.  In  like  manner,  the  osmotic  pressure 
at  the  height  h  +  dh  will  be  f  (n  -  dn)  Ek.  The  difference  in 
osmotic  pressure  between  the  two  levels  is  -f  dnEk  and  since  the 
pressure  acts  over  a  surface  of  s  sq.  cm.,  the  difference  of  osmotic 
forces  acting  over  the  cross-sectional  area  of  the  cylinder  is 
-f  s  dnEk.  Since  the  system  is  in  equilibrium,  this  difference  in 
osmotic  forces  must  be  balanced  by  the  difference  in  the  attraction 


MOLECULAR  REALITY  281 

of  gravitation  at  the  two  levels.  Let  (j>  be  the  volume  of  a  single 
grain,  D  its  density,  and  5  the  density  of  the  suspending  medium. 
The  resultant  downward  pull  upon  a  single  grain  will  be  <£  (D  —  5)  g, 
where  g  is  the  acceleration  due  to  gravity.  The  volume  of  liquid 
between  the  two  levels  being  s  dh,  it  follows  that  the  total  down- 
ward pull  upon  all  the  grains  included  between  the  two  levels  must 
be  nsh(j>  (D  —  5)  g.  It  is  this  force  which  opposes  the  tendency 
of  the  grains  to  distribute  themselves  uniformly  throughout  the 
entire  volume  of  the  suspending  medium,  or,  in  other  words,  it  is 
the  force  which  acts  in  opposition  to  the  osmotic  force  —  §  s  dnEk- 
When  equilibrium  is  established,  these  two  forces  must  be  equal, 
and  we  may  then  write 

-f  s  dnEk  =  ns  dh(j>  (D  -  5)  g.  (3) 

If  n0  and  n  denote  the  number  of  grains  per  unit  of  volume  at  each 
of  two  planes  h  units  apart,  we  obtain,  on  integrating  equation  (3), 

§  Ek  loge  n0/n  =  0  (D  -  5)  gh.  (4) 

On  substituting  in  equation  (4)  the  value  of  Ek  in  equation  (2), 
and  transforming  to  Briggsian  logarithms,  we  have 

2.303  RT/N  log  wo/n  =  |  irr*g  (D  -  5)  h,  (5) 

<£  being  expressed  in  terms  of  the  mean  radius,  r,  of  a  single 
grain.  It  is  evident  that  if  we  can  measure  n,  n0,  D,  and  r  in 
equation  (5),  the  calculation  of  the  Avogadro  Constant,  N,  be- 
comes possible. 

The  determination  of  the  density  of  the  grains,  D,  was  carried 
out  in  two  different  ways  with  suspensions  of  gamboge  and  mastic 
which  had  been  rendered  uniform  by  a  process  of  centrifuging. 
In  the  first  method,  the  grains  were  dried  to  constant  weight 
at  110°,  and  then  by  heating  to  a  higher  temperature,  a  viscous 
liquid  was  obtained  which,  on  cooling,  formed  a  glassy  solid.  The 
density  of  this  solid  was  determined  by  suspending  it  in  a  solution 
of  potassium  bromide  of  known  density. 

In  the  second  method  for  the  determination  of  D,  Perrin  measured 
the  masses  mi  and  m<t  of  equal  volumes  of  water  and  suspension 
respectively.  On  evaporating  the  suspension  to  dryness,  the  mass 
ra3  of  suspended  solid  contained  in  w2  grams  of  suspension  was 


282  THEORETICAL   CHEMISTRY 

obtained.     If  the  density  of  water  is  d,  the  volume  of  the  sus- 
pended grains  will  be 


772-1 


and  consequently  the  density  of  the  grains  will  be  mz/V.  The 
values  of  D  obtained  by  these  two  methods  were  found  to  be  in 
excellent  agreement. 

A  microscope  furnished  with  suitable  micrometers  was  employed 
in  the  determination  of  n  and  n0.  With  the  high  magnification 
employed,  the  depth  of  the  field  of  view  was  limited  :  in  fact,  the 
measurements  were  carried  out  with  a  microscopic  slide  similar 
to  those  used  for  counting  the  corpuscles  in  the  blood.  By  focus- 
sing the  microscope  at  different  depths,  the  average  number  of 
grains  in  the  field  of  view  at  each  level  could  be  counted.  Perrin 
was  able  to  photograph  the  larger  grains  at  different  levels,  whereas 
with  the  smaller  grains  it  was  necessary  to  reduce  the  field  so  that 
relatively  few  grains  were  visible.  The  average  number  of  grains 
counted  at  any  two  different  levels  would  of  course  give  the  de- 
sired ratio  UQ/U. 

The  only  other  quantity  in  equation  (5)  to  be  measured  was  the 
average  radius  of  the  grains  r.  To  determine  this  quantity, 
Perrin  made  use  of  a  method  similar  to  that  used  by  Thomson  for 
counting  the  number  of  electrically  charged  particles  in  an  ionized 
gas.  Stokes  has  shown  that  the  force  required  to  impart  a  uniform 
velocity  v,  to  a  particle  of  radius  r,  moving  through  a  liquid 
medium  whose  viscosity  is  77,  is  given  by  the  formula,  6  irrjrv.  If 
the  motion  be  due  to  gravity,  as  in  the  case  of  suspensions  of  fine 
particles,  obviously  the  foregoing  expression  must  be  equal  to  the 
right-hand  side  of  equation  (5),  or 

6  iryrv  =  §  Trr3  (D  -  d)  g. 

From  this  equation  the  value  of  r  can  be  calculated.  The  rate 
at  which  the  grains  settled  under  the  influence  of  gravity  was 
determined  by  placing  a  portion  of  the  uniform  suspension  in  a 
capillary  tube  and  observing  the  rate  at  which  the  suspension 
cleared,  care  being  taken  to  keep  the  temperature  constant.  This 
method  of  determining  r  was  open  to  the  objection  that  Stokes' 


MOLECULAR  REALITY  283 

law  might  not  apply  to  particles  as  small  as  those  of  colloidal 
suspensions. 

In  order  to  test  the  validity  of  Stokes'  law  under  these  conditions, 
the  following  modification  of  the  method  for  the  determination  of 
r  was  introduced.  It  had  been  observed  that  when  a  suspension  is 
rendered  slightly  acid,  the  grains,  on  coming  in  contact  with  the 
walls  of  the  containing  vessel,  adhered,  while  the  motion  of  the 
grains  throughout  the  bulk  of  the  liquid  remained  unaltered.  In 
this  way  it  was  possible  to  gradually  remove  all  of  the  grains  from 
the  suspension  and  count  them  and,  knowing  the  total  volume  of 
suspension  taken,  the  average  number  of  grains  per  cubic  centi- 
meter could  be  calculated.  If  the  total  mass  of  suspended  matter 
is  known,  it  is  an  easy  matter  to  calculate  the  volume  of  each  grain, 
and  from  this  to  compute  the  radius,  r.  The  value  of  r  determined 
in  this  way  was  found  to  agree  with  that  calculated  by  the  first 
method,  thus  proving  the  validity  of  Stokes'  law  when  applied  to 
colloidal  suspensions. 

Five  series  of  experiments  carried  out  by  Perrin  with  gamboge 
suspensions  in  which  several  thousand  individual  grains  were 
counted,  gave  as  a  mean  value  of  IV  in  equation  (5),  69  X  1022. 
Similar  experiments  with  mastic  suspensions  gave  N  =  70.0  X  1022. 
These  values,  it  will  be  seen,  are  in  close  agreement  with  the  value 
of  Avogadro's  Constant  given  on  page  41. 

The  Law  of  Molecular  Displacement.  The  actual  movements 
of  the  individual  grains  of  a  suspension  when  observed  under  the 
microscope  are  seen  to  be  exceedingly  complex  and  erratic.  The 
horizontal  projections  of  the  paths  of  three  different  grains  in  a 
suspension  of  mastic  are  shown  in  Fig.  72,  the  dots  representing 
the  successive  positions  occupied  by  the  particles  after  intervals  of 
30  seconds.  The  straight  line  joining  the  initial  and  final  positions 
of  a  particle  is  called  the  horizontal  displacement  A,  of  the  particle. 

If  the  time  taken  by  the  particle  to  move  from  its  initial  to  its 
final  position  be  t,  Einstein  *  has  shown  that  the  mean  value  of  the 
square  of  the  horizontal  displacement  of  a  spherical  particle  of 
radius  r  ought  to  be 


N  37r>y 

Zeit.  Elektrochem.,  14,  235  (1908). 


284 


THEORETICAL  CHEMISTRY 


\ 


5 


MOLECULAR  REALITY 


285 


where  y  is  the  viscosity  of  the  suspending  medium  and  where  the 
other  symbols  have  their  usual  significance. 

This  equation  was  tested  by  Perrin,  using  suspensions  of  gam- 
boge and  mastic.  Some  of  the  results  obtained  are  given  in  the 
following  table: 

VALUES  OF  N  CALCULATED  BY  EINSTEIN'S  EQUATION. 


Suspension. 

rin 
microns.* 

TO  X  1015. 

No.  of  dis- 
placements. 

JVxlO-». 

Gamboge  in  water 

0.367 

246 

1500 

69 

Gamboge    in    10%    solution    of 
glycerine 

0.385 

290 

100 

64 

Mastic  in  water.         

0.52 

650 

1000 

73 

Mastic  in  27%  solution  of  urea  .  . 

5.50 

750,000 

100 

78 

*  The  micron  is  one-millionth  of  a  meter  or  one  ten-thousandth  of  a  centimeter. 

It  will  be  seen  that  notwithstanding  the  large  variations  in  the 
granular  masses  of  the  different  suspensions  recorded  in  the  table, 
the  values  of  N,  calculated  by  means  of  Einstein's  equation,  are 
quite  concordant.  Perrin  gives  as  the  mean  value  of  all  of  his 
experiments,  N  =  68.5  X  1022. 

Recent  Investigations  of  the  Brownian  Movement.  Nord- 
lund  *  has  recently  repeated  Perrin 's  experiments,  employing  a 
colloidal  solution  of  mercury  and  an  arrangement  of  apparatus 
whereby  the  movements  of  the  particles  could  be  recorded  photo- 
graphically. The  mean  value  of  N  derived  from  twelve  carefully 
executed  experiments  was  59  X  1022,  the  average  deviation  of  the 
results  of  the  individual  experiments  from  the  mean  being  approxi- 
mately 10  per  cent. 

The  Brownian  Movement  in  gases  has  been  studied  by  Millikan  f 
and  by  Fletcher  t  employing  a  minute  drop  of  oil  as  the  suspended 
particle.  In  the  gaseous  state,  where  the  intermolecular  distances 
are  greater  than  in  the  liquid  state,  not  only  are  the  collisions  less 
frequent  but  the  mean  free  paths  are  appreciably  longer.  These 
conditions  are  favorable  to  the  study  of  the  Brownian  Movement 
and  offer  an  opportunity  for  the  determination  of  the  Avogadro 
Constant  with  a  high  degree  of  accuracy.  As  the  mean  of  nearly 
six  thousand  measurements,  Fletcher  gives  N  =  60.3  X  1022,  this 
value  being  accurate  to  within  1.2  per  cent. 

*  Zeit.  phys.  Chem.,  87,  60  (1914).          f  Phys.  Rev.,  i,  220  (1913). 
$  Ibid.,  4,  453  (1914). 


CHAPTER  XIV. 
THERMOCHEMISTRY. 

General  Introduction.  A  chemical  reaction  is  almost  invari- 
ably accompanied  by  a  thermal  change.  In  the  majority  of 
cases  heat  is  evolved;  a  violent  reaction  developing  a  large  amount 
of  heat,  while  a  feeble  reaction  develops  a  comparatively  small 
amount.  Such  reactions  are  said  to  be  exothermic.  A  relatively 
small  number  of  chemical  reactions  are  known  which  take  place 
with  an  absorption  of  heat.  These  are  termed  endoihermic  reac- 
tions. Instances  of  chemical  reactions  unaccompanied  by  any 
thermal  change  are  very  rare  and  are  almost  wholly  confined  to 
the  reciprocal  transformations  of  optical  isomers.  These  facts, 
which  were  first  observed  by  Boyle  and  Lavoisier,  led  to  the  view 
that  the  amount  of  heat  evolved  in  a  chemical  reaction  might  be 
taken  as  a  measure  of  the  chemical  affinity  of  the  reacting  sub- 
stances. However,  with  the  advance  of  our  theoretical  knowledge, 
it  is  now  known  that  this  is  not  true,  although  a  parallelism 
between  heat  evolution  and  chemical  affinity  frequently  exists. 

Thermochemistry  is  concerned  with  the  thermal  changes  which 
accompany  chemical  reactions. 

Thermal  Units.  Heat  is  a  form  of  energy,  and  like  other 
forms  of  energy  it  may  be  resolved  into  two  factors;  an  intensity 
factor,  the  temperature,  and  a  capacity  factor,  which  may  be 
measured  in  any  one  of  several  units.  Among  these  units  those 
defined  below  are  the  most  frequently  employed. 

The  small  calorie  (cal.)  is  the  quantity  of  heat  required  to  raise 
the  temperature  of  1  gram  of  water  from  15°  C.  to  16°  C.  The 
temperature  interval  is  specified  because  the  specific  heat  of  water 
varies  with  the  temperature.*  The  large  or  kilogram  calorie  (Cal.) 
is  the  quantity  of  heat  required  to  raise  the  temperature  of  1000 
grams  of  water  from  15°  C.  to  16°  C.  The  Ostwald  or  average 

286 


THERMOCHEMISTRY  287 

calorie  (K),  is  the  quantity  of  heat  required  to  raise  the  temper- 
ature of  1  gram  of  water  from  the  melting  point  of  ice  to  the 
boiling  point  of  water  under  a  pressure  of  760  mm.  of  mercury. 
It  is  approximately  equal  to  100  cal.  or  to  0.1  Cal.  The  joule  (j), 
a  unit  based  on  the  C.G.S.  system,  is  equal  to  107  ergs.  This 
being  inconveniently  small  is  generally  multipled  by  1000,  giving 
the  kilojoule  (J),  which  is  therefore  equal  to  1010  ergs.  The  last 
two  units  are  open  to  the  objection  that  their  values  are  depend- 
ent upon  the  mechanical  equivalent  of  heat,  any  change  in  the 
accepted  value  of  which  would  involve  a  correction  of  the  unit  of 
heat.  The  different  capacity  factors  of  heat  energy  are  related 
as  follows :  — 
1  cal.  =  0.001  Cal.  =  0.01  K  (approx.)  =  4.183  j  =  0.004183  J. 

Thermochemical  Equations.  In  order  to  represent  the  changes 
in  energy  which  accompany  chemical  reactions,  an  additional 
meaning  has  been  assigned  to  the  chemical  symbols.  As  ordina- 
rily used,  these  symbols  represent  only  the  molecular  or  formula 
weights  of  the  reacting  substances.  In  a  thermochemical  or 
energy  equation  the  symbols  represent  not  only  the  weight  in 
grams  expressed  by  the  formula  weights  of  the  substances,  but 
also  the  amount  of  heat  energy  contained  in  the  formula  weight 
in  one  state  as  compared  with  the  energy  contained  in  a  standard 
state.  For  example,  the  energy  equation, 

C  +  2  O  =  CO2  +  94,300  cal., 

indicates  that  the  energy  contained  in  12  grams  of  carbon  and 
32  grams  of  oxygen  exceeds  the  energy  contained  in  44  grams  of 
carbon  dioxide,  at  the  same  temperature,  by  94,300  calories.  In 
writing  energy  equations  it  is  very  essential  that  we  have  some 
means  of  distinguishing  between  the  different  states  of  aggrega- 
tion of  the  reacting  substances,  since  the  energy  content  of  a 
substance  is  not  the  same  hi  the  gaseous,  liquid,  and  solid  states. 
In  the  system  proposed  by  Ostwald,  ordinary  type  is  used  for 
liquids,  heavy  type  for  solids,  and  italics  for  gases.  Another  and 
more  convenient  system  has  been  proposed,  in  which  solids  are 
designated  by  enclosing  the  symbol  or  formula  within  square 
brackets;  'liquids  by  the  simple,  unbracketed  symbol  or  formula; 


288 


THEORETICAL  CHEMISTRY 


and  gases  by  enclosing  the  symbol  or  formula  within  parentheses. 
The  above  equation  should,  therefore,  be  written  in  the  following 
manner :  — 

[C]  +  (2  0)  =  (C02)  +  94,300  cal. 

Thermochemical  Measurements.     In  order  to  measure  the 

number  of  calories  evolved  or  ab- 
sorbed when  substances  react,  it  is 
necessary  that  the  reaction  should 
proceed  rapidly  to  completion.  This 
condition  is  fulfilled  by  two  classes 
of  processes.  In  the  first  class  we 
may  mention  the  processes  of  solu- 
tion, hydration,  and  neutralization; 
and  in  the  second  class,  the  process 
of  combustion. 

The  apparatus  used  for  measur- 
ing the  capacity  factor  of  heat  energy 
is  a  calorimeter.  This  instrument 
may  be  given  a  variety  of  forms, 
depending  upon  the  particular  use 
to  which  it  is  to  be  put.  A  simple 
form  of  calorimeter  is  shown  in  Fig. 
73.  It  consists  of  two  concentric 
metal  cylinders,  A  and  B,  insulated 
from  each  other  by  an  air  jacket, 
the  inner  vessel  being  supported 
on  vulcanite  points.  Through  a 
vulcanite  cover  passes  a  thin  walled 
test  tube,  in  which  the  reaction  is 


Fig.  73. 


allowed  to  take  place.  An  accurate  thermometer  and  a  ring-stirrer 
also  pass  through  the  cover  of  the  calorimeter.  In  order  to  deter- 
mine the  thermal  capacity  of  the  calorimeter,  B  is  nearly  filled  with 
water,  and  a  known  mass  of  water,  m,  at  a  temperature  t\  is  intro- 
duced intoC.  Let  the  initial  temperature  of  the  water  in  B  be  tz.  The 
water  in  B  is  stirred  until  the  contents  of  both  B  and  C  have  acquired 
the  same  temperature,  £3.  When  thermal  equilibrium  has  been  estab- 
lished, it  is  evident  that  m  (ti  —  U)  calories  are  required  to  raise 


THERMOCHEMISTRY  289 

the  temperature  of  the  apparatus  and  the  water  in  B,  (^  —  £$) 
degrees.  From  this  data  it  is  an  easy  matter  to  calculate  the 
number  of  calories  required  to  raise  the  temperature  of  the  appar- 
atus and  water  in  B  I  degree,  this  being  the  thermal  capacity  of 
the  apparatus.  The  calorimeter  may  now  be  used  to  determine 
the  heat  evolved  or  absorbed  in  a  reaction.  Suppose,  for  exam- 
ple, that  it  is  desired  to  measure  the  heat  of  neutralization  of  an 
acid  by  a  base.  Equivalent  quantities  of  both  acid  and  base  are 
dissolved  in  equal  volumes  of  water,  care  being  taken  to  make  the 
solutions  dilute.  A  definite  volume  of  one  solution  is  introduced 
into  C  and  an  equal  volume  of  the  other  solution  is  placed  in  a 
vessel  from  which  it  can  be  quickly  and  completely  transferred  to 
C.  When  both  solutions  have  acquired  the  same  temperature, 
the  thermometer  hi  B  is  read  and  then  the  two  solutions  are 
mixed.  When  the  reaction  is  complete,  the  temperature  of  the 
water  in  B  is  again  noted.  If  the  thermal  capacity  of  the  calori- 
meter is  Q,  and  the  rise  in  temperature  produced  by  the  reaction 
is  6,  then  Qd  is  the  amount  of  heat  evolved  by  the  reaction.  To 
this  quantity  of  heat  must  be  added  the  number  of  calories  re- 
quired to  raise  the  temperature  of  the  products  of  the  reaction  8 
degrees.  The  solutions  of  the  products  being  dilute,  their  specific 
heats  may  be  assumed  to  be  equal  to  unity.  From  the  total  quan- 
tity of  heat  so  obtained,  the  number  of  calories  evolved  when  mo- 
lecular quantities  react  can  be  readily  calculated.  The  chief  source 
of  error  in  calorimetric  measurements  is  loss  by  radiation.  This 
may  be  reduced  to  a  minimum  (1)  by  making  the  thermal  capac- 
ity of  the  calorimeter  large,  and  (2)  by  so  arranging  matters  that 
the  initial  temperature  of  the  water  in  the  calorimeter  is  as  much 
below  the  temperature  of  the  room  as  the  final  temperature  is 
above  it. 

The  Combustion  Calorimeter.  The  combustion  of  many  sub- 
stances, such  as  organic  compounds,  proceeds  very  slowly  in  air 
under  ordinary  pressures.  Such  reactions  can  be  accelerated,  if 
they  are  caused  to  take  place  in  an  atmosphere  of  compressed 
oxygen.  For  this  purpose  the  combustion  calorimeter  was  de- 
vised by  Berthelot.*  In  this  apparatus  the  essential  feature  is 
*  Ann.  Chim.  Phys.,  (5),  23,  160  (1881);  (6),  10,  433  (1887). 


290 


THEORETICAL  CHEMISTRY 


the  so-called  combustion  bomb,  shown  in  Fig.  74.  This  consists 
of  a  strong  steel  cylinder  lined  with  platinum  or  gold,  and  fur- 
nished with  a  heavy  threaded  cover.  The  substance  to  be 
burned  is  placed  in  a  platinum  capsule  fastened  to  the  support 
R,  and  a  short  piece  of  fine  iron  wire  of  known  mass  is  connected 
with  the  electric  terminals  Z,  Z,  the  middle  portion  of  the  wire 
dipping  into  the  substance.  The  cover  is  then  screwed  down 
tight,  and  the  bomb  is  filled  with  oxygen  under  a  pressure  of  from 
20  to  25  atmospheres.  The  bomb  is  then 
submerged  in  the  calorimeter,  as  shown  in 
Fig.  75.  The  mass  of  water  in  the  calorim- 
eter being  known  and  its  temperature  having 
I  been  read,  an  electric  current  is  passed  through 
g  S  the  iron  wire  in  the  bomb  causing  it  to  burn 
and  thus  ignite  the  substance.  The  rise  in 
temperature  due  to  the  combustion  is  ob- 
served, and  the  quantity  of  heat  evolved  is 
calculated.  Corrections  must  be  applied  for 
loss  by  radiation,  for  the  heat  evolved  from 
the  combustion  of  the  iron,  and  for  the  heat 
evolved  from  the  oxidation  of  the  nitrogen  of 
the  residual  air  in  the  bomb. 

For  the  details  of  ,the  method  of  determin- 
ing heats  of  combustion  the  student  must 
consult  a  laboratory  manual. 


Fig.  74. 


Law  of  Lavoisier  and  Laplace.  In  1 780,  Lavoisier  and  Laplace,* 
as  a  result  of  their  thermochemical  investigations,  enunciated  the 
following  law :  —  The  quantity  of  heat  which  is  required  to  decompose 
a  chemical  compound  is  precisely  equal  to  that  which  was  evolved  in 
the  formation  of  the  compound  from  its  elements.  This  first  law  of 
thermochemistry  will  be  seen  to  be  a  direct  corollary  of  the  law 
of  the  conservation  of  energy  which  was  first  clearly  stated  by 
Mayer  in  1842. 

Law  of  Constant  Heat  Summation.  A  generalization  of  funda- 
mental importance  to  the  science  of  thermochemistry  was  discov- 
ered in  1840  by  Hess.f  He  pointed  out  that  the  heat  evolved  in  a 

t      *  Oeuvres  de  Lavoisier,  Vol.  II,  p.  283. 
t  Fogg.  Ann.,  50,  385  (1840). 


THERMOCHEMISTRY 


291 


chemical  process  is  the  same  whether  it  takes  place  in  one  or  in  several 
steps.     This  is  known  as  the  law  of  constant  heat  summation.     The 


Fig.  75. 

truth  of  the  law  may  be  illustrated  by  the  equality  of  the  heat  of 
formation  of  ammonium  chloride  in  aqueous  solution,  when  pre- 
pared in  two  different  ways. 


292  THEORETICAL  CHEMISTRY 

Thus, 

(A) 


(NH3)  +  (HC1)  =  [NEUC1]  +42,100  cal. 

[NH4C1]  +  aq.    =  NH4C1,  aq.        -  3,900  cal. 


38,200  cal. 

(B) 

(NH3)  +  aq.  =  NH3,  aq.  +  8,400  cal. 

(HC1)  +  aq.  =  HC1,  aq.  +17,300  cal. 

NH3,  aq.  +  HC1,  aq.  =  NH4Cl,aq.  +12,300  cal. 

38,000  cal. 

It  will  be  observed  that  the  total  amount  of  heat  evolved  in 
the  formation  and  solution  of  ammonium  chloride  is  the  same 
within  the  limits  of  experimental  error,  whether  gaseous  ammonia 
and  hydrochloric  acid  are  allowed  to  react  and  the  resulting  prod- 
uct is  dissolved  in  water,  or  whether  the  gases  are  each  dissolved 
separately  and  then  allowed  to  react.  It  should  be  noted  that 
when  a  substance  is  dissolved  in  so  much  water  that  the  addition 
of  more  water  or  the  removal  of  a  small  portion  of  water  produces 
no  thermal  effect,  it  is  customary  to  denote  it  by  the  symbol  aq. 
(Latin  aqua  =  water).  Thus, 

NH4C1,  aq.  +  nH20  =  NH4C1,  aq., 
NH4C1,  aq.  -  nH2O  =  NH4C1,  aq. 

By  means  of  the  law  of  constant  heat  summation  it  is  possible  to 
find  indirectly  the  amount  of  heat  developed  or  absorbed  by  any 
reaction,  even  though  it  is  impossible  to  carry  it  out  experimen- 
tally. For  example,  it  is  impossible  to  measure  the  heat  evolved 
when  carbon  burns  to  carbon  monoxide.  But  the  heat  evolved 
when  carbon  monoxide  burns  to  carbon  dioxide,  and  also  the  heat 
evolved  when  carbon  burns  to  carbon  dioxide,  can  be  accurately 
determined.  The  energy  equations  are  as  follows :- 

[C]  +  2(0)  =  (C02)  +  94,300  cal.  (1) 

(CO)  +  (0)  =  (C02)  +  67,700  cal.  (2) 


THERMOCHEMISTRY  293 

Treating  these  equations  algebraically,  and  subtracting  equation 
(2)  from  equation  (1),  we  have 

[C]  +  (0)  =  (CO)  +  26,600  caL, 

or,  the  heat  of  combustion  of  carbon  to  carbon  monoxide  is  26,600 
calories.  Again,  as  a  further  illustration  of  the  applicability  of 
the  law  of  Hess,  we  may  take  the  calculation  of  the  heat  of  forma- 
tion of  hydriodic  acid  from  its  elements,  making  use  of  the  follow- 
ing energy  equations :  — 

2  KI,  aq.  +  2  (Cl)  =  2  KC1,  aq.  +  2  [I]  +  524  K  (1) 

2  HI,  aq.  +  2  KOH,  aq.  =  2  KI,  aq.  +  2  H20  +  274  K  (2) 

2  HC1,  aq.  +  2  KOH,  aq.  =  2  KC1,  aq.  +  2  H20  +  274  K  (3) 

2  (HI)  +  aq.  =  2  HI,  aq.  +  384  K,  (4) 

2  (HC1)  +  aq.  =  2  HC1,  aq.  +  346  K,  (5) 

2  (H)  +  2  (Cl)  =  2  (HC1)  +  440  K,  (6) 

adding  equations  (1)  and  (2), 

2  (Cl)  +  2  HI,  aq.  +  2  KOH,  aq.  =  2  KC1,  aq.  +  2  [I]  +  2  H20 

+  798  K.  (7) 

Subtracting  equation  (3)  from  equation  (7), 

2  (Cl)  +  2  HI,  aq.  -  2  HC1,  aq.  =  2  [I]  +  524  K, 
or 

2  (Cl)  +  2  HI,  aq.  =  2  [I]  +  2  HC1,  aq.  +  524  K,  (8) 

adding  equations  (4)  and  (8), 

2  (HI)  +  aq.  +  2  (Cl)  =  2  [I]  +  2  HC1,  aq.  +  908  K,        (9) 
subtracting  equation  (5)  from  equation  (9), 

2  (HI)  +  2  (Cl)  -  2  (HC1)  =  2  [I]  +  562  K, 
or 

2  (HI)  +  2  (Cl)  =  2  [I]  +  2  (HC1)  +  562  K,  (10) 

subtracting  equation  (6)  from  equation  (10), 

2(HI)-2(H)  =2[I]  +  122K, 
or 

2(H)  +  2(I)  =  2  (HI)-  122  K. 

In  a  similar  manner,  practically  any  heat  of  formation  may  be 
calculated,  provided  the  proper  energy  equations  are  combined. 


294  THEORETICAL  CHEMISTRY 

Heat  of  Formation.  The  intrinsic  energy  of  the  substances 
entering  into  chemical  reaction  is  unknown,  the  amount  of  heat 
evolved  or  absorbed  in  the  process  being  simply  a  measure  of  the 
difference  between  the  energy  of  the  reacting  substances  and  the 
energy  of  the  products  of  the  reaction.  Thus,  in  the  equation 

[C]  +  2  (0)  =  (C02)  +  94,300  cal., 

the  difference  between  the  energy  of  a  mixture  of  12  grams  of 
carbon  and  32  grams  of  oxygen,  and  the  energy  of  44  grams  of 
carbon  dioxide  is  seen  to  be  94,300  calories.  The  equation  is 
clearly  incomplete  since  we  have  no  means  of  determining  the 
intrinsic  energies  of  free  carbon  and  oxygen.  Furthermore,  since 
the  elements  are  not  mutually  convertible,  we  have  no  means  of 
determining  the  difference  in  energy  between  them.  It  is  cus- 
tomary, therefore,  in  view  of  this  lack  of  knowledge,  to  put  the 
intrinsic  energies  of  the  elements  equal  to  zero. 

If  the  heats  of  formation  of  the  substances  present  in  a  reac- 
tion are  known,  it  is  much  simpler  to  substitute  these  in  the 
energy  equation  and  solve  for  the  unknown  term.  This  method 
avoids  the  laborious  process  of  elimination  from  a  large  number  of 
energy  equations,  as  in  the  preceding  pages.  If  all  of  the  sub- 
stances involved  in  a  reaction  are  considered  as  decomposed  into 
their  elements,  it  is  evident  that  the  final  result  of  the  reaction 
will  be  the  difference  in  the  sums  of  the  heats  of  formation  on  the 
two  sides  of  the  equation.  This  leads  to  the  following  rule:  — 
To  find  the  quantity  of  heat  evolved  or  absorbed  in  a  chemical  reac- 
tion, subtract  the  sum  of  the  heats  of  formation  of  the  substances 
initially  present  from  the  sum  of  the  heats  of  formation  of  the  products 
of  the  reaction,  placing  the  heat  of  formation  of  all  elements  equal  to 
zero. 

The  energy  equation  for  the  formation  of  carbon  dioxide  from 
its  elements  may  then  be  written  as  follows :  — 

0  +  0  =  (CO2)  +  94,300  cal., 
or 

(CO2)  =  -  94,300  cal. 

That  is,  the  energy  of  1  mol  of  carbon  dioxide  is  —94,300  calories. 
Therefore  in  writing  an  energy  equation  we  make  use  of  the  fol- 


THERMOCHEMISTRY  295 

lowing  rule :  —  Replace  the  formulas  of  each  compound  in  the  equa- 
tion representing  the  reaction  by  the  negative  values  of  their  respective 
heats  of  formation  and  solve  for  the  unknown  term.  This  unknown 
term  may  be  either  the  heat  of  a  reaction  or  the  heat  of  formation 
of  one  of  the  reacting  substances.  The  following  examples  will 
serve  to  illustrate  the  application  of  the  above  rules :  — 

(1)  Let  it  be  required  to  find  the  heat  of  the  following  reaction 

[MgClJ  +  2  [Na]  =  2  [NaCl]  +  [Mg]  +  x, 

where  x  is  the  heat  of  the  reaction.  The  heat  of  formation  of 
MgCl2  is  151  Cal.,  and  that  of  NaCl  is  97.9  Gal,  therefore, 

-  151  +  0  =  -  (2  X  97.9)  +  0  +  z, 
or 

x  =  44.8  Cal. 

(2)  The  heat  of  combustion  of  1  mol  of  methane  is  213.8  Cal., 
and  the  heats  of  formation  of  the  products,  carbon  dioxide  and 
water,  are  94.3  Cal.  and  68.3  Cal.,  respectively.     Let  it  be  required 
to  find  the  heat  of  formation  of  methane.     Representing  the  heat 
of  formation  of  methane  by  x,  we  have 

(CH4)  +  2  (02)  =  (C02)  +  2  (H20)  +  213.8  Cal., 

-  x  +  o  -  -  94.3  -  2  X  69.3  +  213.8, 
or 

x  =  17.1  Cal. 

(3)  The  heat  of  combustion  of  1  mol  of  carbon  disulphide  is 
265.1  Cal.,  the  thermochemical  equation  being 

CS2  +  3  (02)  =  (COa)  +  2  (S02)  +  265.1  Cal. 

The  heats  of  formation  of  carbon  dioxide  and  sulphur  dioxide  are 
94.3  Cal.  and  71  Cal.  respectively.  The  heat  of  formation  of 
carbon  disulphide  x,  may  then  be  calculated  as  follows :  — 

-  x  +  0  =  -  94.3  -  2  X  71  +  265.1, 
or 

x  =  -28.8  Cal. 

Carbon  disulphide  is  thus  seen  to  be  an  endothermic  compound. 

Heat  of  Solution.  The  thermal  change  accompanying  the 
solution  of  1  mol  of  a  substance  in  so  large  a  volume  of  solvent 
that  subsequent  dilution  of  the  solution  causes  no  further  thermal 


296 


THEORETICAL  CHEMISTRY 


change  is  termed  the  heat  of  solution.  The  solution  of  neutral 
salts  is  generally  an  endothermic  process.  This  fact  may  be 
readily  accounted  for  on  the  hypothesis  that  considerable  heat 

HEATS  OF  FORMATION  AND  SOLUTION. 


Substance. 


Heatpf 
Formation. 


Heat  of 
Solution. 


Water,  vapor 58 . 7 

Water,  liquid 68.4 

Hydrochloric  acid 22 . 0 

Sulphuric  acid 193 . 1 

Ammonia 12 . 0 

Nitric  acid 41 . 9 

Phosphoric  acid 302 . 9 

Potassium  hydroxide 103 . 2 

Potassium  chloride 104.3 

Potassium  bromide 95 . 1 

Potassium  iodide 80 . 1 

Potassium  nitrate 119.5 

Sodium  hydroxide 101 . 9 

Sodium  chloride 97 . 6 

Sodium  bromide 85 . 6 

Sodium  sulphate 328.8 

Sodium  nitrate 111.3 

Sodium  carbonate 272 . 6 

Ammonium  chloride 75 . 8 

Ammonium  nitrate 88.0 

Calcium  hydroxide 215.0 

Calcium  chloride 170.0 

Magnesium  sulphate 502 . 0 

Ferrous  chloride 82 . 0 

Ferric  chloride 96 . 1 

Zinc  chloride 97.0 

Zinc  sulphate 30.0 

Cadmium  chloride 293 . 2 

Cupric  chloride 51 . 6 

Cupric  sulphate 182 . 6 

Mercuric  chloride 53 . 2 

Silver  nitrate 28 . 7 

Stannous  chloride 80 . 8 

Stannic  chloride 127 . 3 

Lead  chloride 82.8 

Lead  nitrate..  105.5 


20.3 

17.8 

8.4 

7.2 

2.7 

13.3 

-3.1 

-5.1 

-5.1 

-8.5 

10.9 

1.2 

-0.2 

0.2 

-5.0 

5.6 

-4.0 

-6.2 

3.0 

17.4 

20.3 

17.9 

63.3 

15.6 

18.5 

3.0 

11.1 

15.8 

-3.3 

-5.4 

0.3 

29.9 

-6.8 

-7.6 


must  be  absorbed  as  heat  of  fusion  and  heat  of  vaporization  before 
the  solid  salt  can  assume  a  condition  in  solution  which  closely 
resembles  that  of  a  gas.  The  heat  of  solution  of  hydrated  salts 
is  less  than  the  heat  of  solution  of  the  corresponding  anhydrous 


THERMOCHEMISTRY 


297 


salts.  For  example,  the  heat  of  solution  of  1  mol  of  anhydrous 
calcium  nitrate  is  4000  calories,  while  the  heat  of  solution  of  1  mol 
of  the  tetrahydrate  is  —7600  calories.  The  difference  between 
the  heats  of  solution  of  the  anhydrous  and  hydrated  salts  is  termed 
the  heat  of  hydration.  The  heats  of  formation  and  heats  of  solu- 
tion in  water  of  some  of  the  more  common  compounds  are  given 
in  the  preceding  table,  the  values  being  expressed  in  large  calories. 

Heat  of  Dilution.  The  heat  of  dilution  of  a  solution  is  the 
quantity  of  heat  per  mol  of  solute  which  is  evolved  or  absorbed 
when  the  solution  is  greatly  diluted.  Beyond  a  certain  dilution, 
further  addition  of  solvent  produces  no  thermal  change.  While 
there  is  a  definite  heat  of  solution  for  a  particular  solute  in  a  par- 
ticular solvent,  the  heat  of  dilution  remains  indefinite,  since  the 
latter  is  dependent  upon  the  degree  of  dilution.  Those  gases 
which  obey  Henry's  law  are  practically  the  only  substances  which 
have  no  appreciable  heats  of  solution  or  dilution. 

The  following  tables  give  the  heats  of  dilution  of  hydrochloric 
and  nitric  acids. 

HEAT  OF  DILUTION  OF  SOLUTIONS  OF  HYDROCHLORIC 

ACID. 

Heat  of  solution  =  20.3  cal. 


HC1+H2O 

5.37 

HC1+2  H2O                     

11.36 

HC1  +  10H2O              ; 

16.16 

HC1+50H2O  
HCl-f  300  H2O 

17.1 
17  3 

HEAT  OF  DILUTION  OF  SOLUTIONS  OF  NITRIC  ACID. 
Heat  of  solution  =  7.15  cal. 


HNO3+H2O   

3.84 

HNOs+2  H2O   

2.32 

HNOs+4  H2O  

1.42 

HNOs+6  H2O 

0.2 

HNOa+  8  H2O                 .    .   . 

-0.04 

HNOa+100  H2O    

-0.03 

298  THEORETICAL  CHEMISTRY 

Reactions  at  Constant  Volume.  When  a  chemical  reaction 
takes  place  without  any  change  in  volume,  or  when  the  external 
work  resulting  from  a  change  in  volume  is  not  included  in  the 
heat  of  the  reaction,  the  process  is  said  to  take  place  at  constant 
volume.  That  is  to  say,  the  condition  of  constant  volume  is  a 
condition  which  involves  no  external  work,  either  positive  or 
negative.  Under  these  conditions  the  total  energy  of  the  react- 
ing substances  is  equal  to  the  total  energy  of  the  products  of  the 
reaction,  plus  the  quantity  of  heat  developed  by  the  reaction. 

Reactions  at  Constant  Pressure.  When  a  chemical  reaction 
is  accompanied  by  a  change  in  volume,  the  system  suffers  a  loss 
of  heat  equivalent  to  the  work  done  against  the  atmosphere,  if 
the  volume  increases;  or  the  system  gains  an  amount  of  heat 
equivalent  to  the  work  done  on  the  system  by  the  atmosphere, 
if  the  volume  decreases.  Under  these  conditions  the  reaction  is 
said  to  take  place  at  constant  pressure.  The  difference  between 
constant  volume  and  constant  pressure  conditions,  then,  is  that 
under  the  former,  the  heat  equivalent  of  the  work  corresponding 
to  any  change  in  volume  which  may  occur  is  not  considered  as 
having  any  effect  upon  the  energy  of  the  system;  whereas  under 
the  latter,  due  account  is  taken  of  the  change  in  energy  resulting 
from  change  in  volume.  Suppose  that  in  a  reaction,  1  mol  of  gas 
is  formed.  Under  standard  conditions  of  temperature  and 
pressure  the  volume  of  the  system  will  be  increased  by  22.4  liters. 
The  formation  of  gas  involves  the  performance  of  work  against 
the  atmosphere,  this  work  being  done  at  the  expense  of  the  heat 
energy  of  the  system.  To  calculate  the  heat  equivalent  of  the 
work  done,  let  us  imagine  the  gas  enclosed  in  a  cylinder  fitted  with 
a  piston  whose  area  is  1  square  centimeter.  The  normal  pressure 
of  the  atmosphere  on  the  piston  is  76  cm.  of  mercury  or  1033.3 
grams  per  square  centimeter.  If  the  increase  in  the  volume  of 
the  gas  is  22.4  liters,  the  piston  will  be  raised  through  22,400  cm. 
and  the  work  done  will  be  1033.3  X  22,400  gram-centimeters. 
The  heat  equivalent  of  this  change  in  volume  will  be  (1033.3  X 
22,400)  -J-  42,600  =  542.3  calories  or  0.5423  large  calories.  This 
amount  of  heat  must  be  added  to  the  heat  of  the  reaction.  It 
should  be  observed  that  this  correction  is  independent  of  the  actual 


THERMOCHEMISTRY  299 

value  of  the  pressure  upon  the  system.  Thus,  if  the  pressure  is 
increased  n  times,  the  volume  of  the  gas  will  be  reduced  to  l/n 
of  its  former  value,  and  the  work  done  will  involve  moving  the 
piston  through  l/n  of  the  distance  against  an  n-fold  pressure, 
which  is  plainly  equivalent  to  the  former  amount  of  work.  While 
the  correction  is  independent  of  the  pressure  it  is  not  independent 
of  the  temperature.  The  familiar  equation,  pv  =  RT,  shows  us 
that  the  work  done  by  a  gas  is  directly  proportional  to  its  absolute 
temperature.  Thus,  if  a  gas  is  evolved  at  273°  absolute,  it  will 
occupy  -double  the  volume  it  would  occupy  at  0°,  and  the  work 
done  at  273°  will  involve  moving  the  piston  through  twice  the 
distance  that  it  would  have  to  be  moved  at  0°.  Theoretically,  a 
gas  evolved  at  the  absolute  zero  would  occupy  no  volume  and 
hence  no  work  would  be  done.  Introducing  the  correction  for 
temperature,  we  see  that 

-  X  T  =  1.986  T  cal., 

must  be  added  to  the  heat  of  the  reaction,  where  T  is  the  absolute 
temperature  at  which  the  change  in  volume  occurs.  For  all 
ordinary  purposes  it  is  sufficiently  accurate  to  take  2  T  calories 
as  the  correction.  Thus,  suppose  n  mols  of  gas  to  be  formed  in  a 
reaction  at  17°  C.,  the  amount  of  heat  absorbed  will  be 

nX  2(273  +  17)  =  580  n  cal. 

Under  constant  pressure  conditions,  the  symbols,  in  addition  to 
their  usual  significance,  represent  the  energy  plus  or  minus  the 
term,  2  T  per  mol,  the  positive  or  negative  sign  being  used 
according  as  the  gas  is  absorbed  or  formed.  Since  the  constant 
volume  condition  is  a  condition  in  which  no  account  is  taken  of  the 
external  work,  even  if  a  change  in  volume  does  occur  during  the 
reaction,  and  the  constant  pressure  condition  is  one  in  which 
the  external  work  is  taken  into  consideration,  it  is  apparent  that 
the  relation  of  the  heat  energy  of  a  reaction  at  constant  volume, 
QV)  to  the  heat  energy  at  constant  pressure  Qp,  can  be  represented 
by  the  equation 

cal., 


300  THEORETICAL  CHEMISTRY 

where  n  denotes  the  number  of  mols  of  gas  formed  in  excess  of 
those  initially  present.  This  equation  is  of  great  importance  in 
connection  with  the  determination  of  heats  of  combustion  in  the 
bomb-calorimeter  in  which  the  reactions  necessarily  take  place 
under  constant  volume  conditions.  Since  it  is  customary  to  state 
heats  of  reaction  under  constant  pressure  conditions,  the  foregoing 
equation  makes  it  possible  to  convert  heats  of  combustion  deter- 
mined under  constant  volume  conditions  into  heats  of  combustion 
under  constant  pressure  conditions.  For  example,  the  combustion 
of  naphthalene  takes  place  in  accordance  with  the  equation 
CioH8  +  12  (02)  =  10  (C02)  +  4  (H20)  +  1242.95  Cal. 

It  is  apparent  that  the  combustion  is  accompanied  by  the  forma- 
tion of  2  mols  of  gas,  and  at  15°  C.  the  correction  will  be 

Qp  =  1242.95  -  2  X  0.002  (273  +  15), 
or 

Qp  =  1241.8  Cal. 

The  volume  occupied  by  solids  or  liquids  is  so  small  as  to  be 
negligible  and  does  not  enter  into  these  calculations. 

Variation  of  Heat  of  Reaction  with  Temperature.  If  a  chem- 
ical reaction  be  allowed  to  take  place  first  at  the  temperature  t\, 
and  then  at  the  temperature  ^,  the  amounts  of  heat  developed  in 
the  two  cases  will  be  found  to  be  quite  different.  Let  Qi  and  Q2 
represent  the  quantities  of  heat  evolved  at  the  temperatures  t\ 
and  k,  respectively.  Let  us  imagine  that  the  reaction  takes  place 
at  the  temperature  t\,  Qi  units  of  heat  being  evolved;  and  then 
let  the  products  of  the  reaction  be  heated  to  the  temperature  ^. 
If  c'  represents  the  total  thermal  capacity  of  the  products  of  the 
reaction,  then  the  quantity  of  heat  necessary  to  produce  this  rise 
in  temperature  will  be  c'  (k  —  ti).  Now  let  us  imagine  the 
reacting  substances,  at  the  temperature  t\t  to  be  heated  to  the 
temperature  k,  and  then  allowed  to  react  with  the  evolution  of 
Q2  units  of  heat.  The  heat  necessary  to  produce  this  rise  in 
temperature  in  the  reacting  substances  is  c  (^  —  £1),  where  c 
is  the  total  thermal  capacity  of  the  original  substances.  Having 
started  with  the  same  substances  at  the  same  initial  temperature, 
and  having  obtained  the  same  products  at  the  same  final  temper- 


THERMOCHEMISTRY  301 

ature,  we  have,  according  to  the  law  of  the  conservation  of 
energy, 

Qi  -  c'  (fe  -  *0  =  Q2  -  c  &  -  *0, 
or 


or,  where  the  change  in  temperature  is  very  small, 

^-c-c' 
dt  "°      C' 

If  c'  is  greater  than  c  then  the  sign  of  dQ/dt  will  be  negative,  or,  in 
other  words,  an  increase  in  temperature  will  cause  a  decrease  in 
the  heat  of  reaction.  On  the  other  hand,  if  c  is  greater  than  c', 
dQ/dt  will  be  positive  and  the  heat  of  reaction  will  increase  with 
the  temperature. 

EXAMPLE.     The   reaction   between   hydrogen  and   oxygen  at 
18°  C.  is  represented  by  the  following  equation^  — 

2  (H2)  +  (O2)  =  2  (H20)  +  1367.1  K. 

Suppose  it  is  required  to  find  how  much  heat  will  be  evolved  when 
equal  masses  of  the  two  gases  react  at  110°  C.,  the  product  of  the 
reaction  being  maintained  at  this  temperature,  and  the  pressure 
remaining  constant.  The  specific  heats  per  gram  of  the  different 
substances  involved  are  as  follows  :  — 

Hydrogen   =  3.409;    Oxygen   =  0.2175;    Water  (between  18° 
and    100°)  =  1;   Water    (between    100°   and    110°)  =0.5. 
The  heat  of  vaporization  of  water  is  537  calories  per  gram. 

For  liquid  water  per  degree  we  have, 
dQ/dt  =  (4  X  3.409  +  32  X  0.2175)  -  (36  X  1)  =  -  15.404  cal. 

and  for  (100°-  18°)  =82°,  we  have,  82  X  (-15.404)  =-1263  cal. 
The  heat  of  formation  of  liquid  water  at  100°  is,  therefore, 
1367.1  -  12.63  =  1354.47  K. 

When  the  liquid  water  is  vaporized  at  100°,  (36  X  537)  calories  of 
this  heat  is  absorbed,  or  the  formation  of  steam  at  100°  from 
hydrogen  and  oxygen,  evolves 

1354.47  -  193.32  =  1161.15  K. 


302 


THEORETICAL  CHEMISTRY 


For  steam  per  degree,  we  have, 

dQ/dt  =  (4  X  3.409  +  32  X  0.2175)  -  (36  X  0.5)  =  2.596  cal., 
and  for  the  interval  (110°  ~-  100°)  =  10°, 

10  X  2.596  =  25.96  cal. 
Or  for  the  total  heat  evolved,  we  have 

1161.15  +  0.2596  -  1161.41  K. 

Heat  of  Combustion.  The  heat  evolved  during  the  complete 
oxidation  of  unit  mass  of  a  substance  is  termed  its  heat  of 
combustion.  The  unit  of  mass  commonly  chosen  in  all  physico- 
chemical  calculations  is  the  mol.  An  enormous  amount  of 
experimental  work  has  been  done  by  Thomsen,*  Berthelot,t  and 
Langbein  t  on  the  determination  of  the  heats  of  combustion  of  a 
large  number  of  organic  compounds.  A  few  of  their  results  are 
given  in  the  accompanying  tables. 

SATURATED  HYDROCARBONS. 


Hydrocarbon. 

Heat  of 
Combustion. 

Difference. 

Methane,  CH4  

Cal. 
211   9 

Cal. 

Ethane,  C2H6  

370  4 

158.5 

Propane  CsHg 

529  2 

158.8 

Butane,  C4Hio  

687.2 

158.0 

Pentane,  CsHi2 

847  1 

159.9 

UNSATURATED  HYDROCARBONS. 


Hydrocarbon. 

Heat  of 
Combustion. 

Difference. 

Ethylene,  C2H4  

Cal. 

333  4 

Cal. 

Propylene,  C3H6  

492  7 

159.3 

Isobutylene,  C4H8  

650.6 

157.9 

Amylene    CsHio 

807  6 

157.0 

Acetylene,  CaH2  

310.1 

Allylene,  C3H4  

467.6 

157.5 

*  Thermochemische  Untersuchungen,  4  Vols. 

t  Essai  de  Mecanique  Chimique,  Thermochimie,  Donees  et  Lois  Numeri- 
ques. 

$  Jour,  prakt.  Chem.,  1885  to  1895. 


THERMOCHEMISTRY  303 

« 

ALCOHOLS. 


Alcohol. 

Heat  of 
Combustion. 

Difference. 

Methyl  alcohol,  CH4O 

Cal. 
182  2 

Cal. 

Ethyl  alcohol,  C2H6O    .  . 

340  5 

158.3 

Propyl  alcohol,  C3H8O  

498  6 

158.1 

Isobutyl  alcohol,  C4Hi0O  

658  5 

159.9 

It  will  be  observed  that  a  very  nearly  constant  difference  in 
the  heat  of  combustion  corresponds  to  a  constant  difference  of  a 
CH2  group  in  composition.  A  number  of  interesting  relations 
between  heats  of  combustion  of  compounds  and  their  differences 
in  composition  have  been  discovered,  but  these  cannot  be  taken  up 
at  this  time.  It  has  also  been  pointed  out  that  the  heat  of  com- 
bustion of  organic  compounds  is  conditioned  not  only  by  their 
composition,  but  also  by  their  molecular  constitution. 

Some  exceedingly  interesting  and  important  results  have  been 
obtained  with  the  different  allotropic  forms  of  the  elements.  For 
example,  when  equal  masses  of  the  three  common  allotropic  forms 
of  carbon  are  burned  in  oxygen,  the  amounts  of  heat  evolved  are 
found  to  be  quite  different,  as  is  shown  by  the  following  energy 
equations :  — 

[C]  diamond  +  2  (0)  =  (C02)  •  +  94.3  Cal. 
[C]  graphite  +  2  (O)  =  (C02)  +  94.8  Cal. 
[C]  amorphous  +  2  (0)  =  (C02)  +  97.65  Cal. 

It  is  apparent  that  amorphous  carbon  contains  the  greatest 
amount  of  energy  of  any  one  of  the  three  allotropic  modifications, 
and,  .therefore,  when  amorphous  carbon  is  changed  into  diamond, 
the  reaction  must  be  accompanied  by  the  evolution  of  (97.65  — 
94.3)  =  3.35  Cal.  In  like  manner,  the  allotropic  forms  of  sulphur 
and  phosphorus  have  different  heats  of  combustion.  The  follow- 
ing equations  show  the  heat  equivalents  of  the  differences  in 
intrinsic  energy  between  the  allotropic  forms :  — 

S  (monoclinic)  =  S  (rhombic)  +  2.3  Cal. 
P  (white)  =  P  (red)  +  3.71  Cal. 


304  THEORETICAL  CHEMISTRY 

When  the  same  substance  is  burned  in  oxygen  and  then  in  ozone, 
it  is  found  that  more  heat  is  evolved  in  ozone  than  in  oxygen. 
The  energy  equation  expressing  the  change  of  ozone  into  oxygen 
may  be  written  thus, 

(08)  =  1 J  (02)  +  36.2  Cal. 

All  of  the  above  facts  illustrate  the  general  principle  that  the 
larger  amounts  of  intrinsic  energy  are  associated  with  the  more 
unstable  forms. 

Thermoneutrality  of  Salt  Solutions.  In  addition  to  the  law 
of  constant  heat  summation,  Hess  discovered  two  other  important 
laws  of  thermochemistry,  viz.,  the  law  of  thermoneutrality  of 
salt  solutions,  and  the  law  governing  the  neutralization  of  acids 
by  bases.*  When  two  dilute  salt  solutions  are  mixed  there  is 
neither  evolution  nor  absorption  of  heat.  Thus  when  dilute  solu- 
tions of  sodium  nitrate  and  potassium  chloride  are  mixed,  there  is 
no  thermal  effect.  The  energy  equation  may  be  written  as  follows  :— 

NaN03,  aq.  +  KC1,  aq.  =  NaCl,  aq.  +  KNO3,  aq.  +  0  Cal. 

According  to  this  equation  a  double  decomposition  has  taken 
place  and  we  should  naturally  expect  an  evolution  or  an  absorption 
of  heat.  While  Hess  could  not  account  for  the  absence  of  any 
thermal  effect,  he  recognized  the  fact  as  quite  general  and  formu- 
lated the  law  of  the  thermoneutrality  of  salt  solutions  as  fol- 
lows:—  The  metathesis  of  neutral  salts  in  dilute  solutions  takes  place 
with  neither  evolution  nor  absorption  of  heat. 

The  explanation  of  the  phenomenon  of  thermoneutrality  was 
furnished  by  the  theory  of  electrolytic  dissociation.  When  the 
above  equation  is  written  in  the  ionic  form,  it  becomes 

Na  +  N08'  +  K*  +  Cl'  =  Na*  +  Cl'  +  K*  +  NO,'. 

From  this  it  is  apparent  that  the  same  ions  exist  on  both  sides  of 
the  equation,  and  in  reality  no  reaction  takes  place. 

There   are   numerous    exceptions   to   the    law  of    thermoneu- 
trality.    These  can  be  satisfactorily  accounted  for  by  the  theory  of 
electrolytic  dissociation.     All  of  those  salts,  the  behavior  of  which  in 
dilute  solution  is  contrary  to  the  law,  are  found  to  be  only  partially 
*  Pogg.  Ann.,  50,  385  (1840). 


THERMOCHEMISTRY  305 

ionized,  and,  therefore,  when  their  solutions  are  mixed,  a  chem- 
ical reaction  actually  occurs.  The  exceptions  must  be  considered 
as  furnishing  additional  evidence  in  favor  of  the  theory  of  elec- 
trolytic dissociation. 

Heat  of  Neutralization.  Hess  also  discovered  *  that  when 
dilute  solutions  of  equivalent  quantities  of  strong  acids  and 
strong  bases  are  mixed,  practically  the  same  amount  of  heat  is 
evolved.  The  following  energy  equations  may  be  considered  as 
typical  examples  of  such  neutralizations :  — 

HC1,  aq.  +  NaOH,  aq.  =  NaCl,  aq.  +  H2O  +  13.75  Cal., 
HN03,  aq.  +  NaOH,  aq.  =  NaN03  aq.  +  H2O  +  13.68  Cal., 
HC1,  aq.  +  KOH,  aq.  =  KC1,  aq.  +  H2O  +  13.70  Cal., 
HN03,  aq.  +  KOH,  aq.  =  KN03,  aq.  +  H2O  +  13.77  Cal., 
HC1,  aq.  +  LiOH,  aq.  =  LiCl,  aq.  +  H2O  +  13.70  Cal. 

Here  again  it  would  be  difficult  to  explain  the  phenomenon  with- 
out the  theory  of  electrolytic  dissociation.  In  terms  of  this 
theory,  however,  the  explanation  is  perfectly  plausible.  If  MOH 
and  HA  represent  any  strong  base  and  any  strong  acid  respectively, 
then  when  equivalent  amounts  of  these  are  dissolved  in  water, 
each  solution  being  largely  diluted  to  the  same  volume,  the  reac- 
tion may  be  written  thus :  — 

M'  +  OH'  +  H-  +  A'  =  M-  +  A'  +  H20  +  13.7  Cal. 

Disregarding  the  ions  which  occur  on  both  sides  of  the  equality 
sign,  we  have 

OH'  +  ET  =  H20  +  13.7  Cal. 

It  thus  appears  that  the  neutralization  of  a  strong  acid  by  a  strong 
base  in  dilute  solution  consists  solely  in  the  combination  of  hydro- 
gen and  hydroxyl  ions  to  form  undissociated  water,  the  heat  of 
this  ionic  reaction  being  13.7  large  calories. 

The  heat  of  formation  of  water  from  its  ions  must  not  be  con- 
fused with  the  heat  of  formation  of  water  from  its  elements. 
When  weak  acids  or  weak  bases  are  neutralized  by  strong  bases 
or  strong  acids,  or  when  weak  acids  are  neutralized  by  weak  bases, 

*  Loc.  cit. 


306  0  THEORETICAL  CHEMISTRY 

the  heat  of  neutralization  may  differ  widely  from  13.7  Cal.  This 
is  shown  by  the  following  thermochemical  equations :  — 

H-COOH,  aq.  +  NaOH,  aq.  =  H-COONa,  aq.  +  H20 

+  13.40  Cal., 
HC12-COOH,  aq.  +  NaOH,  aq.  =  HCl2-COONa,  aq.  +  H20 

+  14.83  Cal., 
H-COOH,  aq.  +  NH4OH,  aq.  =  H-COONH4,  aq.  +  H20 

+  11.90  Cal., 
HCN,  aq.  +  NaOH,  aq.  =  NaCN,  aq.  +  H2O  +  2.90  Cal. 

As  will  be  seen,  the  heat  of  neutralization  may  be  either  greater 
or  less  than  13.7  Cal.  The  exceptions  to  the  generalization  of 
constant  heat  of  neutralization  are  readily  explained  by  the 
theory  of  electrolytic  dissociation.  Suppose  a  weak  acid  to  be 
neutralized  by  a  strong  base.  According  to  the  dissociation 
theory,  the  acid  is  only  slightly  dissociated  and,  therefore,  yields 
a  comparatively  small  number  of  hydrogen  ions  to  the  solution. 
The  base  on  the  other  hand  is  completely  dissociated  into  hydroxyl 
and  metallic  ions.  Therefore,  as  many  hydroxyl  ions  disappear 
as  there  are  free  hydrogen  ions  with  which  they  can  combine  to 
form  water.  When  the  equilibrium  between  the  acid  and  the 
products  of  its  dissociation  has  been  thus  disturbed,  it  undergoes 
further  dissociation  and  the  resulting  hydrogen  ions  immediately 
combine  with  the  free  hydroxyl  ions  of  the  base.  This  process 
continues  until  all  of  the  hydroxyl  ions  of  the  base  have  been 
neutralized.  It  is  evident  that  the  thermal  effect  in  this  case  is 
the  algebraic  sum  of  the  heat  of  dissociation  of  the  weak  acid, 
which  may  be  positive  or  negative,  and  the  heat  of  formation  of 
water  from  its  ions.  A  similar  explanation  holds  for  the  neutrali- 
zation of  a  weak  base  by  a  strong  acid,  or  for  the  neutralization 
of  a  weak  acid  by  a  weak  base.  This  affords  a  method  for  estimat- 
ing the  approximate  value  of  the  heat  of  dissociation  of  a  weak 
acid  or  a  weak  base.  For  example,  in  the  equation  given  above, 

HCN,  aq.  +  NaOH,  aq.  =  NaCN,  aq.  +  H2O  +  2.90  Cal., 

the  difference  between  2.9  and  13.7  or  —10.8  Cal.  represents 
approximately  the  heat  of  dissociation  of  hydrocyanic  acid. 


THERMOCHEMISTRY 


307 


Since  the  acid  is  initially  slightly  dissociated  in  dilute  solution, 
it  is  apparent  that  in  order  to  obtain  the  true  heat  of  dissociation, 
we  must  add  to  — 10.8  Cal.  the  thermal  value  of  the  dissociation 
of  that  portion  of  the  acid  which  has  already  become  ionized. 
Heat  of  lonization.  Since  13.7  Cal.  is  the  heat  of  formation 
of  water  from  its  ions,  this  must  also  be  the  thermal  equivalent  of 
the  energy  required  to  dissociate  one  mol  of  water  into  its  ions.  It 
must  be  remembered  that  the  dissociated  molecule  of  water  must 
be  mixed  with  a  very  large  volume  of  undissociated  water,  in  order 

HEAT  OF  FORMATION  OF  IONS. 


Ion. 

Heat  of 
Formation. 

Ion. 

Heat  of 
Formation. 

Hydrogen 

0.0 

Copper  (ic)  

-15.8 

Potassium 

61.9 

Copper  (ous)  

-16.0 

Sodium                         .    . 

57.5 

Mercury  (ous)  

-19.8 

Lithium 

62  9 

Silver 

—25  3 

Ammonium 

32  8 

Lead 

0  5 

IVIagnesium 

109  0 

Tin  (ous) 

3.3 

Calcium 

109  0 

Chlorine     .               ... 

39.3 

Aluminium  
Manganese 

121.0 
50.2 

Bromine  
Iodine  

28.2 
13.1 

Iron  (ous). 

22.2 

Sulphate  

214.4 

Iron  (ic) 

-9.3 

Sulphite  

151.3 

Cobalt. 

17.0 

Nitrous  

27.0 

Nickel.                       .    .  . 

16.0 

Nitric  

49.0 

Zinc 

35  1 

Carbonate 

161.1 

Cadmium 

18.4 

Hyroxyl  ... 

54.7 

that  the  dissociation  may  be  permanent.  Reference  to  the  table 
of  heats  of  formation  (p.  296),  will  show  that  68.4  Cal.  are  required 
to  form  one  mol  of  water  from  its  elements.  Hence,  it  follows  that 
68.4  —  13.7  =  54.7  Cal.,  is  the  heat  of  formation  of  one  equivalent 
of  hydrogen  and  hydroxyl  ions.  It  has  been  shown  that  an  ex- 
tremely small  amount  of  energy  is  necessary  to  ionize  hydrogen 
when  it  is  dissolved  in  water.  It  is  evident,  therefore,  that  54.7 
Cal.  is  a  close  approximation  to  the  heat  of  formation  of  one  equiva- 
lent of  hydroxyl  ions. 

On  the  assumption  that  the  heat  of  ionization  of  gaseous  hydro- 
gen in  solution  is  zero,  the  values  of  the  other  ionic  heats  of  forma- 


308  THEORETICAL  CHEMISTRY 

tion  may  be  computed.  For  example,  the  heat  of  formation  of 
KOH,  aq.  is  116.5  Cal.  The  ionic  heat  of  formation  of  potassium 
ions  must  be  116.5  —  54.7  =  61.8  Cal.  In  like  manner,  the 
heat  of  formation  of  KC1,  aq.  is  101.2  Cal.;  hence  the  ionic  heat 
of  formation  of  chlorine  ions  must  be  101.2  —  61.8  =  39.4  Cal. 
The  preceding  table  of  ionic  heats  of  formation  has  been  calculated 
as  in  the  above  examples. 

The  Principle  of  Maximum  Work.  A  fundamental  principle 
of  the  science  of  mechanics  is,  that  a  system  is  in  stable  equilib- 
rium when  its  potential  energy  is  a  minimum.  In  1879,  Ber- 
thelot  *  suggested  that  a  similar  principle  applies  to  chemical 
systems. 

In  terms  of  the  kinetic  theory,  the  temperature  of  a  substance 
is  to  be  regarded  as  a  measure  of  the  kinetic  energy  of  its  molecules. 
The  development  of  heat  by  a  chemical  reaction  would,  therefore, 
be  taken  as  an  indication  of  a  decrease  in  the  potential  energy  of 
the  system.  Berthelot's  theorem,  known  as  the  principle  of 
maximum  work,  may  be  stated  as  follows:  —  "Every  chemical  proc- 
ess accomplished  without  the  intervention  of  any  external  energy 
tends  to  produce  that  substance  or  system  of  substances  which  evolves 
the  maximum  amount  of  heat."  The  table  of  heats  of  formation 
(p.  296),  illustrates  the  general  truth  of  this  principle,  but  as  will 
be  seen,  the  theorem  precludes  the  possibility  of  spontaneous 
endothermic  reactions.  Thus,  for  example,  the  formation  of 
acetylene  from  its  elements  at  the  temperature  of  the  electric 
arc  is  a  well-known  endothermic  reaction,  but  according  to  the 
principle  of  maximum  work,  it  could  not  take  place  spontaneously. 
Another  serious  objection  to  Berthelot's  principle  is,  that  accord- 
ing to  it,  all  chemical  reactions  should  proceed  to  completion,  the 
reaction  taking  place  in  such  a  way  as  to  evolve  the  greatest  amount 
of  heat.  As  is  well  known,  many  reactions,  and  theoretically  all 
reactions,  are  never  complete,  but  proceed  until  a  condition  of 
equilibrium  is  reached.  The  principle  of  maximum  work,  there- 
fore, denies  the  existence  of  equilibria  in  chemical  reactions. 
Many  attempts  have  been  made  to  "explain  away"  these  defects, 
but  none  of  them  have  been  successful.  In  referring  to  the  generali- 
*  Essai  de  Mecanique  Chimique. 


THERMOCHEMISTRY  309 

zation,  Le  Chatelier  terms  it  "a  very  interesting  approximation 
toward  a  strictly  valid  generalization. " 

The  Theorem  of  Le  Chatelier.  As  a  result  of  his  attempts  to 
modify  the  principle  of  maximum  work  and  render  it  generally 
applicable,  Le  Chatelier  was  led  to  the  discovery  of  a  rigorous  law 
of  wide-reaching  usefulness.  His  generalization  may  be  stated 
as  follows:  — •  Any  alteration  in  the  factors  which  determine  an  equi- 
librium) causes  the  equilibrium  to  become  displaced  in  such  a  way 
as  to  oppose,  as  far  as  possible,  the  effect  of  the  alteration.  If  the 
temperature  of  a  system  which  is  in  equilibrium  be  raised  or 
lowered,  the  resulting  displacement  of  the  equilibrium  is  accom- 
panied by  such  absorption  or  evolution  of  heat  as  will  tend  to 
maintain  the  temperature  constant.  An  interesting  illustration 
of  the  behavior  of  a  system  when  one  of  the  factors  controlling 
the  equilibrium  is  varied,  is  afforded  by  the  system 

2N02<=»N2O4. 

The  reaction  proceeds  in  the  direction  indicated  by  the  upper 
arrow  with  the  evolution  of  12.6  Cal.  Increase  of  temperature 
favors  the  reaction  which  is  accompanied  by  an  absorption  of 
heat,  which  in  this  case,  is  the  reaction  indicated  by  the  lower 
arrow.  Hence  as  the  temperature  rises,  the  percentage  of  NO2 
increases  at  the  expense  of  N2O4.  This  fact  can  be  demonstrated 
by  the  following  experiment.  Some  liquefied  N204  is  placed  in 
each  of  three  long  glass  tubes,  which  are  sealed  at  one  end.  When 
enough  N2O4  has  vaporized  to  displace  the  air,  the  open  ends  of 
the  tubes  are  sealed.  Changes  in  the  equilibrium  caused  by 
varying  the  temperature  can  be  followed  by  noting  the  changes 
in  the  color  of  the  mixture.  N204  is  an  almost  colorless  substance, 
while  N02  is  reddish  brown.  At  ordinary  temperatures  the 
contents  of  the  tubes  will  .be  brown  in  color.  One  tube  is  set 
aside  as  a  standard  of  comparison,  while  the  temperature  of  the 
second  is  lowered  by  surrounding  it  with  a  freezing  mixture.  As 
the  temperature  falls,  the  brown  color  of  the  contents  of  the  tube 
becomes  much  lighter,  showing  an  increased  formation  of  NoOi- 
The  third  tube  is  heated  by  immersing  it  in  a  beaker  of  boiling 
water.  As  the  temperature  rises,  the  contents  of  the  tube  becomes 


310  THEORETICAL  CHEMISTRY 

much  darker  in  color,  indicating  an  increase  in  the  amount  of  N(>2 
in  the  mixture. 

Another  example  is  afforded  by  the  equilibrium  between  ozone 
and  oxygen,  represented  by  the  equation 

2O3<=±3O2. 

The  reaction  indicated  by  the  upper  arrow  is  exothermic.  In- 
crease of  temperature  causes  a  displacement  of  the  equilibrium 
in  the  direction  of  the  lower  arrow,  since  under  these  conditions 
heat  is  absorbed.  Thus,  as  the  temperature  rises  ozone  becomes 
increasingly  stable.  Nernst  has  calculated  that  at  6000°  C., 
the  temperature  of  the  photosphere  of  the  sun,  10  per  cent  of  the 
above  equilibrium  mixture  would  be  ozone.  Other  applications 
of  the  theorem  of  Le  Chatelier  will  be  given  in  subsequent 
chapters. 

PROBLEMS. 

1.  From  the  following  data  calculate  the  heat  of  formation  of  HN02 
aq.- 

[NH4N02]  =  (N2)  +  2  H20  +  71.77  Gal., 

2  (H2)  +  (02)  =  2  H20  +  136.72  Gal., 

(N2)  +  3  (H2)  +  aq.  =  2  NH3  aq.  +  40.64  Gal., 

NH3aq.  +  HN02aq.  =  NH4N02aq.  +  9.110  Gal., 

[NH4N02]  +  aq.  =  NH4N02aq.  -  4.75  Cal. 

Am.    (H)  +  (N)  +  (02)  +  aq.  =  HN02  aq.  +  30.77  Cal. 

2.  By  the  combustion  at  constant  pressure  of  2  grams  of  hydrogen 
with  oxygen  to  form  liquid  water  at  17°  C.,  68.36  Cal.  are  evolved.     What 
is  the  heat  evolution  at  constant  volume?  Ans.   67.49  Cal. 

3.  The  heats  of  solution  of  Na2S04,  Na2S04.H20,  and  Na2S04.10H2O 
are  0.46,   -  1.9  and  -  18.76  Cal.  respectively.     What  are  the  heats  of 
hydration  of  Na2S04;   (a)  to  monohydrate,  (b)  to  decahydrate? 

Ans.    (a)  2.36  Cal.,  (b)  19.22  Cal. 

4.  The  heats  of  neutralization  of  NaOH  and  NH4OH  by  HC1  are  13.68 
and  12.27  Cal.  respectively.    What  is  the  heat  of  ionization  of  NH4OH, 
if  it  is  assumed  to  be  practically  undissociated?  Ans.   1.41  Cal. 


THERMOCHEMISTRY  311 

5.  From  the  following  energy  equations:  — 

[C]  +  (02)  =  (C02)  +  96.96  Cal., 

2  (H2)  +  (02)  =  2  H20  +  136.72  Cal., 

2  C6H6  +  15  (02)  =  12  (C02)  +  6  H20  +  1598.7  Cal., 

2  (C2H2)  +  5  (02)  =  4  (C02)  +  2  H20  +  620.1  Cal., 

all  at  17°  C.  and  constant  pressure,  calculate  the  heat  evolved  at  17°  C. 
in  the  reaction 

3  C2H2  =  Cell  e, 

(a)  at  constant  pressure,  and  (b)  at  constant  volume. 

Ans.    (a)  130.8  Cal.,  (b)  129.07  Cal. 

6.  Calculate  the  heat  of  formation  of  sulphur  trioxide  from  the  follow- 
ing energy  equations:  — 

[PbO]  +  [S]  +  3  (0)  =  [PbS04]  +  1655  K. 
[PbO]  +  H2S04.5  H20  =  [PbS04]  +  6  H20  +  233  K. 
[S]  +  3  (0)  +  6  H20  =  H2S04.5  H20  +  1422  K. 
[S03]  +  6  H20  =  H2S04.5  H20  +  411  K. 

Ans.   [S]  +  3  (0)  =  [S03]  +  1011  K. 

7.  What  is  the  heat  of  formation  of  a  very  dilute  solution  of  calcium 
chloride?     (See  table  on  p.  257.)  Ans.   187.6  Cal. 


CHAPTER  XV. 
HOMOGENEOUS   EQUILIBRIUM. 

Historical  Introduction.  In  this  and  the  two  succeeding 
chapters,  the  conditions  which  affect  the  rate  and  the  extent  of 
chemical  reactions  will  be  considered.  When  two  substances 
react  chemically,  it  is  customary  to  refer  the  phenomenon  to  the 
existence  of  an  attractive  force  known  as  chemical  affinity. 

Ever  since  the  metaphysical  speculations  of  the  Greeks,  who 
endowed  the  atoms  with  the  instincts  of  love  and  hate,  the  nature 
of  chemical  affinity  has  been  under  discussion.  So  little  has  been 
learned  as  to  the  cause  of  chemical  reactions,  that  in  recent  years 
this  question  has  been  dismissed  and  attention  has  been  directed 
to  the  more  promising  question  as  to  how  they  take  place.  New- 
ton's discovery  of  the  law  of  gravitation  led  him  to  consider  the 
attraction  between  atoms  and  the  attraction  between  large  masses 
of  matter  as  manifestations  of  the  same  force. 

Although  Newton  found  that  chemical  attraction  does  not 
follow  the  law  of  the  inverse  square,  yet  his  suggestion  exerted 
a  profound  influence  upon  the  minds  of  his  contemporaries. 

Geoffrey  and  Bergmann  arranged  chemical  substances  in  the 
order  of  their  displacing  power.  Thus,  if  we  have  three  sub- 
stances, A,  J5,  and  C  and  the  attraction  between  A  and  B  is 
greater  than  that  between  A  and  C,  then  when  B  is  added  to  AC 
it  will  completely  displace  C,  as  indicated  by  the  following  equa- 
tion:- AC  +  B  =  AB  +  C. 

These  investigators  overlooked  a  factor  of  fundamental  importance 
in  conditioning  chemical  reactivity,  viz.,  the  influence  of  mass. 
The  importance  of  the  relative  amounts  of  the  reacting  substances 
in  determining  the  course  of  a  reaction  was  first  clearly  recognized 
by  Wenzel  *  in  1777.  It  remained  for  Berthollet,t  however,  to 

*  Lehre  von  der  chemischen  Verwandtschaft  der  Korper. 
t  Essai  de  Statique  Chimique. 
312 


HOMOGENEOUS  EQUILIBRIUM  313 

point  out  the  significance  of  the  views  advanced  by  Wenzel. 
His  first  paper  on  this  subject  was  published  in  1799,  while  acting 
as  a  scientific  adviser  to  Napoleon  on  his  Egyptian  expedition. 
Under  ordinary  conditions  sodium  carbonate  and  calcium  chloride 
react  according  to  the  equation, 

NasCOs  +  CaCl2  =  2  NaCl  +  CaC03, 

the  reaction  proceeding  nearly  to  completion.  Berthollet  observed 
the  deposits  of  sodium  carbonate  on  the  shores  of  certain  saline 
lakes  in  Egypt,  and  pointed  out  that  this  salt  is  produced  by  the 
reversal  of  the  above  reaction,  the  large  excess  of  sodium  chloride 
in  solution  in  the  water  of  the  lakes  conditioning  the  course  of 
the  reaction. 

The  German  chemist  Rose*  furnished  much  additional  evidence 
in  favor  of  the  effect  of  mass  on  chemical  reactions.  He  pointed 
out  that  in  nature,  the  silicates,  which  are  among  the  most  stable 
compounds  known,  are  undergoing  a  continual  decomposition 
under  the  influence  of  such  relatively  weak  agents  as  water  and 
carbon  dioxide.  The  relatively  strong  specific  affinities  of  the 
atoms  of  the  silicates  are  overcome  by  the  preponderating  masses 
of  water  and  carbon  dioxide  in  the  atmosphere.  In  1862  an 
important  contribution  to  our  knowledge  of  the  effect  of  mass  on 
the  course  of  a  chemical  reaction  was  made  by  Berthelot  and  Pean 
de  St.  Gilles.f  They  investigated  the  formation  of  esters  from 
alcohols  and  acids.  The  reaction  between  ethyl  alcohol  and 
acetic  acid  is  represented  by  the  equation 

C2H5OH  +  CH3COOH  <=±  CH3COOC2H5  +  H20. 

Starting  with  equivalent  quantities  of  alcohol  and  acid,  the  reac- 
tion proceeds  until  about  two-thirds  of  the  reacting  substances 
have  been  converted  into  ester  and  water.  In  like  manner,  if 
equivalent  quantities  of  ethyl  acetate  and  water  are  brought 
together,  the  reaction  proceeds  in  the  direction  indicated  by  the 
lower  arrow,  until  about  one-third  of  the  original  substances  have 
been  converted  into  acid  and  alcohol.  In  other  words  the  reac- 
tion is  reversible,  a  condition  of  equilibrium  resulting  when  the 

*  Pogg.  Ann.,  94,  481  (1855);  95,  96,  284,  426  (1855). 

t  Ann.  Chim.  Phys.  [3],  65,  385;  66,  5;  68,  225  (1862-1863). 


314 


THEORETICAL  CHEMISTRY 


speeds  of  the  two  reactions,  indicated  by  the  upper  and  lower 
arrows,  become  equal.  If  now  a  fixed  amount  of  acid  is  taken, 
say  1  equivalent,  and  the  quantity  of  alcohol  is  varied,  a  corre- 
sponding displacement  of  the  equilibrium  follows. 

The  following  table  gives  the  results  obtained  by  Berthelot  and 
Pean  de  St.  Gilles  for  ethyl  alcohol  and  acetic  acid.  The  first 
and  third  columns  give  the  number  of  equivalents  of  alcohol  to 
1  equivalent  of  acetic  acid,  and  the  second  and  fourth  columns 
give  the  percentage  of  ester  formed. 


Equivalents 
of  Alcohol. 

Ester 
Formed. 

Equivalents 
of  Alcohol. 

Ester 
Formed. 

0.2 

19.3 

2.0 

82.8 

0.5 

42.0 

4.0 

88.2 

1.0 

66.5 

12.0 

93.2 

1.5 

77.9 

50.0 

100.0 

The  effect  of  increasing  the  mass  of  alcohol  on  the  course  of  the 
reaction  is  very  beautifully  shown  by  the  above  results. 

The  Law  of  Mass  Action.  While  the  influence  of  the  relative 
,  masses  of  the  reacting  substances  in  conditioning  chemical  reac- 
tions was  thus  fully  established,  it  was  not  until  1867  that  the  law 
governing  the  action  of  mass  was  accurately  formulated. 

In  that  year  Guldberg  and  Waage,*  two  Scandinavian  investiga- 
tors, enunciated  the  law  of  mass  action  as  follows: —  The  amount 
of  chemical  action  at  any  stage  of  a  reaction  is  proportional  to  the 
active  masses  of  the  reacting  substances  present  at  that  time.  Guld- 
berg and  Waage  defined  the  term  "active  mass"  as  the  molecular 
concentration  of  the  reacting  substances.  It  is  to  be  carefully 
noted  that  the  amount  of  chemical  action  is  not  porportional 
to  the  actual  masses  of  the  substances  present,  but  rather  to  the 
amounts  present  in  unit  volume.  The  law  is  generally  applicable 
to  homogeneous  systems;  that  is,  to  those  systems  in  which 
ordinary  observation  fails  to  reveal  the  presence  of  essentially 
different  parts.  The  amount  of  chemical  action  exerted  by  a 
*  Etudes  sur  les  Affinite*s  Chimiques,  Jour,  prakt.  Chem.  [2],  19,  69  (1879). 


HOMOGENEOUS  EQUILIBRIUM  315 

substance  can  be  determined,  either  from  its  effect  on  the  equili- 
brium, or  from  its  influence  on  the  speed  of  reaction. 

In  order  to  apply  the  law  of  mass  action  practically,  it  must  be 
formulated  mathematically.  Let  a  and  b  denote  the  molecular 
concentrations  of  the  substances  initially  present  in  a  reversible 
reaction.  According  to  the  law  of  mass  action,  the  rate  at  which 
these  substances  combine  is  proportional  to  the  active  masses 
of  each  constituent,  and  therefore  to  their  product,  ah.  The 
initial  speed  of  the  reaction  at  the  time  to  is  therefore, 

Speed<0  oo  ab,  or  Speed^  =  k  •  ab, 

in  which  the  proportionality  factor  fc,  is  known  as  the  velocity 
constant.  As  the  reaction  proceeds,  the  molecular  concentrations 
of  the  original  substances  steadily  diminish,  while  the  molecular 
concentrations  of  the  products  of  the  reaction  steadily  increase. 
Let  us  assume  that  after  the  interval  of  time  t,  x  equivalents  of 
the  products  of  the  reaction  have  been  formed.  The  speed  of 
the  original  reaction  will  now  be 

Speedf  =  k  (a  —  x)  (b  —  x). 

As  the  reaction  proceeds,  the  tendency  of  the  products  to  combine 
and  reform  the  original  substances  increases.  At  the  time  t, 
when  the  concentration  of  the  products  is  x,  the  speed  of  the 
reverse  reaction  will  be 

Speed*  =  ki  •  xz, 

where  ki  is  the  velocity  constant  of  the  reverse  reaction. 

We  thus  have  two  reactions  proceeding  in  opposite  directions: 
the  speed  of  the  direct  reaction  continuously  diminishes  while 
that  of  the  reverse  reaction  continually  increases.  It  is  evident 
that  a  point  must  ultimately  be  reached  at  which  the  speeds  of 
the  direct  and  reverse  reactions  become  equal,  and  a  condition 
of  equilibrium  will  be  established.  Let  x±  represent  the  value  of 
x  when  equilibrium  is  attained;  we  then  have 

Speed  direct  =  k  (a  —  Xi)  (b  —  Xi)  =  Speed  reverse 

or 

(a  -  xi)  (b  -  xi)      /bi       v 
_ —  __,  =  /< 

Xi2  k 


316  THEORETICAL  CHEMISTRY 

in  which  K  is  known  as  the  equilibrium  constant.  Since  the  veloc- 
ity constants  k  and  k\t  are  independent  of  the  concentration,  it 
follows  that  the  above  equation  holds  for  all  concentrations. 
Therefore,  if  the  value  of  the  equilibrium  constant  of  a  reaction 
is  known,  the  equilibrium  conditions  can  be  calculated  for  any 
concentrations  of  the  reacting  substances.  When  more  than  one 
mol  of  a  substance  is  involved  in  a  reaction,  each  mol  must  be 
considered  separately  in  the  mass  action  equation. 
Thus,  let 

niAi  +  nzAz-\-  .  .  .  +±ni'Ai  4-  nz'Az  +  .  .  . 

represent  any  reversible  reaction,  hi  which  HI  mols  of  A\  and  n^ 
mols  of  A2  react  to  form  n\  mols  of  A\  and  nz'  mols  of  A2f.  When 
equilibrium  is  attained,  we  shall  have 


or 

c%c£  •  •  •   -h-K  m 

<xc".'  '  fc  -Ac> 

CAl'CA1'    '    '    • 

in  which  the  symbol  c  is  used  to  denote  the  active  mass,  or  molecular 
concentration  of  the  substances  involved  in  the  reaction.  This 
is  a  perfectly  general  form  of  the  mass-action  equation.  Since  at 
any  one  temperature,  concentration  and  pressure  are  proportional, 
we  may  write  equation  (1)  in  the  following  form 


which,  in  the  case  of  gaseous  equilibria,  is  often  a  more  convenient 
form  of  the  equation. 
The  relation  between  the  two  equilibrium  constants,  Kc  and  Kp, 

can  be  easily  determined,  as  follows:  —  Since  c  =  -  =  -^m   we 
have,  on  substituting  this  value  of  c  in  equation  (1), 

Y* 

RT) 


T      \RT 


HOMOGENEOUS  EQUILIBRIUM  317 

or  indicating  the  sum  of  the  initial  number  of  mols  by  I>n  and  the 
sum  of  the  final  number  of  mols  by  Z??',  we  have 
KC  =  KP(RT)  *»'-*". 

It  is  evident,  therefore,  that  in  reactions  where  the  same  number 
of  mols  occur  on  both  sides  of  the  equality  sign,  Kc  =  KP.  Equa- 
tion (1)  (or  equation  (2))  is  sometimes  known  as  the  reaction 
isotherm.  While  the  law  of  mass  action  may  be  proved  thermody- 
namically,  a  much  simpler  kinetic  derivation  has  been  given  by 
van't  Hoff.  If  we  assume  that  the  rate  of  chemical  change  is  pro- 
portional to  the  number  of  collisions  per  unit  of  time  between  the 
molecules  of  the  reacting  substances,  then  in  the  reaction 

niAi  +  n2A2  +  .  .  .  =  ni'Ai  +  nz'Az  +  ...:, 
the  velocity  of  the  direct  change  will  be  kc^c^  .  .  .  and  the 
velocity  of  the  reverse  reaction  will  be  kic^,^,.  .  .  . 

At  equilibrium,  the  two  velocities  will  be  equal  and,  therefore, 


or 

CAfA,    '    '    '  fcl         ^ 

——, — —, =  -r~  =  A. 

ft\         W2  L* 

c    c 

As  a  consequence  of  the  assumptions  involved  in  both  the  thermo- 
dynamic  and  the  kinetic  proofs  of  the  law  of  mass  action,  it  fol- 
lows that  the  law  is  only  strictly  applicable  to  very  dilute  solutions. 
Notwithstanding  this  limitation,  experimental  results  indicate 
that  it  frequently  holds  for  moderately-concentrated  solutions. 

Equilibrium  in  Homogeneous  Gaseous  Systems. 

(a)  Decomposition  of  Hydriodic  Acid.  A  typical  example  of 
equilibrium  in  a  gaseous  system  is  afforded  by  the  decomposition 
of  hydriodic  acid,  as  represented  by  the  equation 

This  reaction  has  been  thoroughly  investigated  by  Hautefeuille, 
Lemoine  and  Bodenstein.*  The  reaction  is  well  adapted  for 
investigation  since  it  proceeds  very  slowly  at  ordinary  temper- 

*  Zeit.  Phys.  Chem.  22,  1  (1897). 


318  THEORETICAL  CHEMISTRY 

atures,  while  at  the  temperature  of  boiling  sulphur,  448°  C., 
equilibrium  is  established  quite  rapidly.  If  the  mixture  of  gases 
is  maintained  at  448°  C.  for  some  time  and  is  then  cooled  quickly, 
the  respective  concentrations  of  the  components  of  the  mixture 
can  be  determined  by  the  ordinary  methods  of  chemical  analysis. 
Various  mixtures  of  the  gases  are  sealed  in  glass  tubes  and  heated 
for  a  definite  time  in  the  vapor  of  boiling  sulphur.  The  tubes 
are  then  cooled  rapidly  to  the  temperature  of  the  room  and,  after 
the  iodine  and  hydriodic  acid  have  been  removed  by  absorption  in 
potassium  hydroxide,  the  amount  of  free  hydrogen  present  in  each 
tube  is  measured. 

Applying  the  law  of  mass  action  to  the  above  equation,  we  have 


CHI 

Expressing  the  analytical  results  in  mols,  let  a  mols  of  iodine  be 
mixed  with  b  mols  of  hydrogen,  and  let  2  x  mols  of  hydriodic  acid 
be  formed.  Then  when  equilibrium  is  established,  a  —  x  will  be 
the  amount  of  iodine  vapor  and  b  —  x  will  be  the  amount  of  hydro- 
gen present.  The  concentrations  being  directly  proportional  to 
the  amounts  present,  we  may  substitute  these  values  for  c#2,  c/2, 
and  CHI  hi  the  mass-action  equation.  The  following  expression 
is  thus  obtained :  — 

(6  -  z)  (a  -  z) 

=  A/.. 


Solving  the  equation  for  x,  we  obtain 


a  +  b  -  Va2  +  62  -  ab  (2  -  16  Kc) 


Since,  according  to  Avogadro's  law,  equal  volumes  of  all  gases 
contain  the  same  number  of  molecules,  volumes  may  be  sub- 
stituted for  a,  6,  and  x.  Bodenstein  expressed  his  results  in  terms 
of  volumes  reduced  to  standard  conditions  of  temperature  and 
pressure.  On  analyzing  equilibrium  mixtures,  Bodenstein  found 
that  at  448°  C.,  Kc  =  0.01984,  and  at  350°  C.,  Kc  =  0.01494. 

Having  determined  the  value  of  the  equilibrium  constant,  he 
made  use  of  this  value  in  calculating  the  volume  of  hydriodic 


HOMOGENEOUS  EQUILIBRIUM 


319 


acid  which  should  t  be  obtained  from  known  volumes  of  hydrogen 
and  iodine.  A  comparison  of  the  calculated  and  observed  values 
showed  excellent  agreement.  The  following  table  contains  a  few 
of  the  results  obtained  by  Bodenstein  at  448°  C. 


Hydrogen, 
b. 

Iodine, 
a. 

HI  calculated, 
2x. 

HI  observed, 
2z. 

20.57 

5.22 

10.19 

10.22 

20.60 

14.45 

25.54 

25.72 

15.75 

11.90 

20.65 

20.70 

14.47 

38.93 

27.77 

27.64 

8.10 

2.94 

5.64 

5.66 

8.07 

9.27 

13.47 

13.34 

It  is  of  interest  to  note  that  a  change  in  pressure  does  not 
alter  the  equilibrium  in  this  gaseous  system.  Making  use  of  the 
partial  pressures  of  the  components  of  the  gaseous  system  instead 
of  the  concentrations,  we  have 

Pff2  *  Ph  _  j? 

rf        ~     p' 
PHI 

Now  let  the  total  pressure  on  the  system  be  increased  to  n  times 
its  original  value;  then  the  partial  pressures  are  all  increased  in 
the  same  proportion,  and  we  have 


which  is  equivalent  to  the  original  expression,  since  n  cancels 
out.  The  equilibrium  is  thus  seen  to  be  independent  of  the  pres- 
sure. This  is  only  true  for  those  systems  in  which  a  change  in 
volume  does  not  occur. 

(b)  Dissociation  of  Phosphorus  Pentachloride.  When  phos- 
phorus pentachloride  is  vaporized  it  dissociates  according  to  the 
following  equation 


Applying  the  law  of  mass  action,  we  have 


320  THEORETICAL  CHEMISTRY 

Starting  with  1  mol  of  phosphorus  pentachloride,  which  if  undis- 
sociated  would  occupy  the  volume  V,  under  atmospheric  pressure, 
and  letting  a  denote  the  degree  of  dissociation,  the  molecular  con- 
centrations at  equilibrium  will  be  as  follows:  — 


Cpa'  = 


Letting  (1  +  a)  V  =  V,  and  substituting  in  the  above  equation, 
we  have 


At  250°  C.  phosphorus  pentachloride  is  dissociated  to  the  extent 
of  80  per  cent.     Under  atmospheric  pressure  1  mol  will  be  present 

273  4-  250 
in  22.4 "—liters  =  V.     The  final  volume  will,  therefore,  be 

ZiO 

V  =  (1  +  0.8)  (22.4  2732|325°) 

The  value  of  the  equilibrium  constant,  —  usually  designated  in 
cases  of  dissociation,  the  dissociation  constant,  —  is,  therefore, 

(0.8)2 


(1-0.8)  (1+0.8)  ^22.4 


273  +  250V 
273      / 


Having  obtained  the  value  of  KC)  the  direction  and  extent  of  the 
reaction  at  250°  C.  can  be  determined,  provided  the  initial  molec- 
ular concentrations  are  known.  The  reaction  is  accompanied  by 
a  change  in  volume,  and,  therefore,  the  equilibrium  is  displaced  by 
a  change  in  pressure.  Making  use  of  the  partial  pressures  of  the 
components  of  the  gaseous  mixture,  we  have 


where  pi  and  pz  are  the  partial  pressures  of  phosphorus  penta- 
chloride and  the  products  of  the  dissociation,  phosphorus  tri- 


HOMOGENEOUS  EQUILIBRIUM  321 

chloride  and  chlorine,  respectively.     Let  the  total  pressure  be 
increased  n-times,  then 


pi 

It  is  apparent  from  this  equation,  that  the  equilibrium  is  not 
independent  of  the  pressure,  an  increase  in  pressure  being  accom- 
panied by  a  diminution  of  the  dissociation.  An  important  point 
in  connection  with  dissociation,  first  observed  by  Deville,*  is  the 
effect  on  the  equilibrium  of  the  addition  of  an  excess  of  one  of  the 
products  of  dissociation.  For  example,  in  the  equilibrium 


an  excess  of  chlorine  or  of  phosphorus  trichloride,  drives  back  the 
dissociation.  If  pi  denotes  the  partial  pressure  of  phosphorus 
pentachloride,  p2  that  of  phosphorus  trichloride,  and  ps  that  of 
chlorine,  then  we  have 

Pz-ps  _  K 

-   —  -A-p. 
Pi 

Now  let  an  excess  of  chlorine  be  added;  this  will  cause  the  value 
of  p3  to  increase.  Since  the  value  of  Kp  is  constant,  the  value  of 
pz  must  diminish  and  that  of  p\  must  increase.  Hence,  the  addi- 
tion of  an  excess  of  either  product  of  dissociation  causes  a  diminu- 
tion of  the  amount  of  the  dissociation. 

(c)  Dissociation  of  Carbon  Dioxide.     Carbon  dioxide  dissociates 
according  to  the  equation, 


This  is  a  somewhat  more  complex  gaseous  system  than  either  of 
the  foregoing  systems.  When  equilibrium  is  established,  let 
pi  be  the  partial  pressure  of  the  carbon  dioxide,  p2  the  partial  pres- 
sure of  carbon  monoxide,  and  pz  the  partial  pressure  of  oxygen, 
then  we  have 


At  3000°  C.  and  under  atmospheric  pressure,  carbon  dioxide  is 
*  Le§ons  sur  la  dissociation,  Paris  (1866). 


322  THEORETICAL  CHEMISTRY 

40  per  cent  dissociated.     The  partial  pressures  of  each  of  the  com- 
ponents may  be  readily  calculated  as  follows:  — 

2(1-0.40)  _ 

2  (1  -  0.40)  +  3  X  0.40 

2  X  0.40 

2  (1  -  0.40)  +  3  X  0.40 
_  0.40 
P*  ~  2  (1  -  0.40)  +  3  X  0.40  ~ 

Substituting  these  values  in  the  above  equation,  we  obtain 

_  (0.33)'  X  0.17 
p  (0.50)2 

The  dissociation  constant  for  carbon  dioxide  may  have  a  different 
value  if  the  equation  is  written  in  the  form 


Applying  the  law  of  mass  action,  we  have 


Substituting  the  above  values  of  the  partial  pressures,  we  obtain 

Kp  =  0.272. 

Equilibrium  in  Liquid  Systems.     The  reaction  between  an 
alcohol  and  an  acid  to  form  an  ester  and  water  may  be  taken  as 
an  example  of  equilibrium  in  a  liquid  system.     In  the  reaction 
C2H5OH  +  CHsCOOH^  CH3COOC2H5  +  H20, 

let  a,  6,  and  c  represent  the  number  of  mols  of  alcohol,  acid  and 
water  respectively,  which  are  present  in  V  liters  of  the  mixture, 
and  let  x  denote  the  number  of  mols  of  ester  and  water  which 
have  been  formed  when  the  system  has  reached  equilibrium. 
The  active  masses  of  the  components  will  then  be, 

n      -  a  ~  x      r       -  b  ~  x  •     r        -x-      AT         _c  +  x 

^alc.  ~~         TT      )      ^acid  ~         -rr       j      ^  ester  —  "TT  >  and      U  water  ~        V     ' 

Applying  the  law  of  mass  action,  we  obtain 
(a  -  x)  (b  -  x)  __  „ 
x(c  +  x)        -Ac' 


HOMOGENEOUS  EQUILIBRIUM 


323 


In  this  case  the  value  of  the  equilibrium  constant  is  independent 
of  the  volume.  This  reaction  has  been  studied,  as  already  men- 
tioned, by  Berthelot  and  Pean  de  St.  Gilles.*  They  found  that 
when  equivalent  amounts  of  alcohol  and  acid  are  mixed,  the  reac- 
tion proceeds  until  two-thirds  of  the  mixture  is  changed  into  ester 
and  water.  Hence,  we  find 

ixj 
"I  xf 


Kc 


1. 


Having  determined  the  value  of  Kc,  it  may  now  be  used  to  cal- 
culate the  equilibrium  conditions  for  any  initial  concentrations 
of  the  substances  involved  in  the  reaction.  As  an  illustration, 
we  will  take  1  mol  of  acetic  acid  and  treat  it  with  varying  amounts 
of  alcohol,  the  initial  mixture  containing  neither  of  the  products  of 
the  reaction.  The  equation  takes  the  form 

(a  -  x)  (1  -  x)  _  1 
~~tf~          ~4* 
Solving  for  x,  we  have 

x  =  f  (1  +  a  -  Va2-a  +  l). 

A  comparison  of  the  observed  and  calculated  values  given  in  the 
accompanying  table  shows  that  the  agreement  is  excellent,  even 
in  the  more  concentrated  solutions,  where  we  might  reasonably 
expect  that  the  mass  law  would  cease  to  hold. 


Alcohol, 
a. 

Ester 
(observed  )  , 
z. 

Ester 
(calculated), 
z. 

0.05 

0.05 

0.049 

0.08 

0.078 

0.078 

0.18 

0.171 

0.171 

0.28 

0.226 

0.232 

0.33 

0.293 

0.311 

0.50 

0.414 

0.523 

0.67 

0.519 

0.528 

1.0 

0.665 

0.667 

1.5 

0.819 

0.785 

2.0 

0.858 

0.845 

2.24 

0.876 

0.864 

8.0 

0.966 

0.945 

*  Loc.  cit. 


324  THEORETICAL  CHEMISTRY 

The  Variation  of  the  Equilibrium  Constant  with  Temperature. 
Van't  Hoff  showed  that  the  displacement  of  equilibrium  due  to 
change  in  temperature  is  connected  with  the  heat  evolved  or  ab- 
sorbed in  a  chemical  reaction  by  the  equations 


dT         '  RT2' 
and 

d(logeKp)_  Qp 
dT  RT2' 

where  Qv  and  Qp  are  the  heats  of  reaction  at  constant  volume  and 
constant  pressure  respectively,  and  where  R  and  T  have  their 
usual  significance.  Equation  (1)  is  known  as  the  reaction  isochore. 
Both  equations  show  that  the  rate  of  change  of  the  natural  log- 
arithm of  the  equilibrium  constant  with  temperature  is  equal  to 
the  total  heat  of  reaction  divided  by  the  molecular  gas  constant 
times  the  square  of  the  absolute  temperature  at  which  the 
reaction  takes  place.  Equations  (1)  and  (2)  hold  only  for  displace- 
ments of  the  equilibrium  due  to  infinitely  small  changes  in  temper- 
ature. In  order  to  render  these  equations  applicable  to  concrete 
equilibria,  it  is  necessary  to  integrate  them.  The  integration  of 
these  expressions  can  only  be  performed  if  Q  is  constant.  For 
small  intervals  of  temperature,  Q  is  practically  independent  of 
the  temperature,  and  for  larger  intervals  we  may  take  the  value 
of  Q  which  corresponds  to  the  mean  of  the  two  temperatures 
between  which  the  integration  is  performed.  Integrating  equa- 
tions (1)  and  (2)  on  this  assumption,  we  obtain 


log.  Kct  -  log,  K  Cl  =  f  (jr  -  f9 ) '  (3) 

and 

log.  Kp.  -  log.  K»  =  J  (jr  ~  j$  •  (4) 

Passing  to  Briggsian  logarithms,  and  putting  R  =  1.99  calories, 
equations  (3)  and  (4)  become 

n         /rr          rr  \ 

(5) 


HOMOGENEOUS  EQUILIBRIUM  325 

and 


We  shall  now  proceed  to  show  how  these  important  equations 
may  be  applied  to  several  typical  equilibria. 

(a)  Vaporization  of  Water.     The  equilibrium  between  a  liquid 
and   its  vapor   is   conditioned    by    the   pressure   of  the    vapor, 
this  in  turn  being  dependent  upon  the  temperature.     In  this  case 
of  physical  equilibrium,  we  have  KPI  =  pi,  and  KP2  =  p2.     The 
value  of  Qp  for  water  can  be  calculated  from  the  following  data:  — 

Ti  =  273°,  pi  =    4.54  mm.  of  mercury, 

T2  =  273°  +11°.54,  p2  =  10.02  mm.  of  mercury. 

Substituting  in  equation  (6),  we  have 

4.581  (log  10.02  -  log  4.54)  273  X  284.5 
^p  ~  273  -  284.5 

or  Qp  =  —  10,670  calories. 

The  value  of  Qp  obtained  by  experiment  is  —10,854  calories. 

(b)  Dissociation  of  Nitrogen  Tetroxide.     In  the  reaction 


the  following  values  for  the  dissociation  of  N204  have  been  ob- 

tained: — 

771  =  273°+   26°.l,  «i  =  0.1986, 

T2  =  273°  +  111°.3,  <*2  =  0.9267. 

If  the  dissociation  takes  place  under  a  pressure  of  1  atmosphere, 
then  the  partial  pressures  of  the  component  gases  will  be 

~ 


The  values  of  KPI  and  K^  are,  then,  according  to  the  law  of 
mass  action  as  follows  :  — 


326  THEORETICAL  CHEMISTRY 

and 


Substituting  in  equation  (6)  and  solving  for  Qp,  we  obtain 

.  4  X  (0.9267)2  4XC0.1986)2 

4.581  log  2  -  log  1  _  (Q  2  299.1  X  384.3 


p  299.1  -  384.3 

or 

Qp  =  -  12,260  calories  per  mol  of  N204. 

In  a  reaction  which  is  accompanied  by  no  thermal  change, 
Q  =  0,  and  the  right-hand  side  of  equations  (1)  and  (2)  becomes 
equal  to  zero.  In  other  words,  in  such  a  reaction  a  change  in 
temperature  does  not  cause  a  displacement  of  the  equilibrium. 

The  reaction, 

C2H5OH  +  CH3COOH^±  CH3COO.C2H6  +  H20, 

is  accompanied  by  such  a  small  thermal  change  that  it  may  be 
considered  as  zero,  and  according  to  the  above  reasoning  there 
should  be  only  a  very  slight  displacement  of  the  equilibrium  when 
the  temperature  is  varied.  Berthelot  found  that  at  10°  C., 
65.2  per  cent  of  the  alcohol  and  acid  are  changed  into  ester,  and 
at  220°  C.,  66.5  per  cent  of  the  mixture  is  transformed  into  ester. 
As  will  be  seen,  an  increase  of  210°  produces  hardly  any  displace- 
ment of  the  equilibrium. 

PROBLEMS. 

1.  When  2.94  mols  of  iodine  and  8.10  mols  of  hydrogen  are  heated  at 
constant  volume  at  444°  C.  until  equilibrium  is  established,  5.64  mols 
of  hydriodic  acid  are  formed.     If  we  start  with  5.30  mols  of  iodine  and 
7.94  mols  of  hydrogen,  how  much  hydriodic  acid  is  present  at  equilibrium 
at  the  same  temperature?  Ans.   9.49  mols. 

2.  At  2000°  C.,  and  under  atmospheric  pressure,  carbon  dioxide  is 
1.80  per  cent  dissociated  according  to  the  equation 


Calculate  the  equilibrium  constant  for  the  above  reaction  using  partial 
pressures.  Ans.  3  X  10~6. 


HOMOGENEOUS  EQUILIBRIUM  327 

3.  What  is  the  equilibrium  constant  in  the  preceding  problem,  if  the 
concentrations  are  expressed  in  mols  per  liter?  Ans.   1.61  X  10~8. 

4.  When  6.63  mols  of  amylene  and  1  mol  of  acetic  acid  are  mixed, 
0.838  mol  of  ester  is  formed  in  the  total  volume  of  894  liters.     How  much 
ester  will  be  formed  when  we  start  with  4.48  mols  of  amylene  and  1  mol 
of  acetic  acid  in  the  volume  of  683  liters?  Ans.  0.8111  mol. 

5.  If  1  mol  of  acetic  acid  and  1  mol  of  ethyl  alcohol  are  mixed,  the 
reaction 

C2H5OH  +  CH3COOH  <=±  CH3COOC2H5  +  H20, 

proceeds  until  equilibrium  is  reached,  when  f  mol  of  ethyl  alcohol,  £  mol 
of  acetic  acid,  f  mol  of  ethyl  acetate,  and  f  mol  of  water  are  present.  If 
we  start  (a)  with  1  mol  of  acid  and  2  mols  of  alcohol;  (b)  with  1  mol  of 
acid,  1  mol  of  alcohol,  and  1  mol  of  water;  (c)  with  1  mol  of  ester  and  3 
mols  of  water,  how  much  ester  will  be  present  in  each  case  at  equilibrium? 
Ans.  (a)  0.845  mol,  (b)  0.543  mol,  (c)  0.465  mol. 

6.  In  the  reaction 


we  find,  since  3  0  =  i  02,  for 


the  values  3.02  at  386°  C.  and  2.35  at  419°  C.     Calculate  the  heat  evolved 
by  the  reaction  under  constant  pressure.  Ans.  6827  cal. 

7.   Above  150°  C.  N02  begins  to  dissociate  according  to  the  equation 


At  390°  C.  the  vapor  density  of  N02  is  19.57  (H  =  1),  and  at  490°  C 
it  is  18.04.  Calculate  the  degree  of  dissociation  according  to  the  above 
equation  at  each  of  these  temperatures;  the  equilibrium  constants 
expressing  the  concentrations  in  mols  per  liter;  and  the  heat  of  dissoci- 
ation of  N02. 

Ans.   ai  =  0.35,    «2  =  0.55.     Kl  =  2.884  X  10~2. 
K2  =  7.173  X  10-2.  Q  =  -9407  cal. 


CHAPTER   XVI. 
HETEROGENEOUS  EQUILIBRIUM. 

Heterogeneous  Systems.  We  have  now  to  consider  equilibria 
in  systems  made  up  of  matter  in  different  states  of  aggregation. 
Such  systems  are  termed  heterogeneous  systems,  as  distinguished 
from  those  dealt  with  in  the  preceding  chapter  where  the  compo- 
sition is  uniform  throughout.  The  physically  distinct  portions  of 
matter  involved  in  a  heterogeneous  system  are  known  as  phases, 
each  phase  being  homogeneous  and  separated  from  the  other 
phases  by  definite  bounding  surfaces.  Thus,  ice,  liquid  water 
and  vapor  constitute  a  physically  heterogeneous  system.  Another 
heterogeneous  system  is  formed  by  calcium  carbonate  and  its 
dissociation  products,  calcium  oxide  and  carbon  dioxide.  The 
equilibrium  between  a  solid,  its  saturated  solution,  and  vapor 
affords  an  illustration  of  a  still  more  complex  heterogeneous 
system. 

Application  of  the  Law  of  Mass  Action  to  Heterogeneous 
Equilibria.  It  has  been  shown  in  the  preceding  chapter  that 
the  law  of  mass  action  may  be  applied  to  homogeneous  equilibria 
provided  the  molecular  condition  of  the  reacting  substances  is 
known. 

When  we  attempt  to  apply  the  law  of  mass  action  to  hetero- 
geneous  equilibria,    especially    where    solids    are    involved,    the 
problem  presents  difficulties.     In  his  investigation  of  the  dis- 
sociation of  calcium  carbonate,  according  to  the  equation 
[CaCOJ  <=±  [CaO]  +  (CO2), 

Debray  *  showed  that  just  as  every  liquid  has  a  definite  vapor 
pressure  corresponding  to  a  certain  temperature,  so  there  is  a 
definite  pressure  of  carbon  dioxide  over  calcium  carbonate  at  a 
definite  temperature.  Furthermore,  the  pressure  was  found  to 
be  independent  of  the  amount  of  calcium  carbonate  present. 

*  Compt.  rend.,  64,  603  (1867). 
328 


HETEROGENEOUS  EQUILIBRIUM  329 

Guldberg  and  Waage  *  showed  that  the  law  of  mass  action  can 
be  applied  to  such  heterogeneous  equilibria,  provided  that  the 
active  masses  of  the  solids  present  are  considered  as  constant. 

Nernst  pointed  out  that  this  statement  of  Guldberg  and  Waage 
can  be  easily  reconciled  with  experimental  facts.  In  a  hetero- 
geneous equilibrium  involving  solids,  it  is  only  necessary  to  con- 
sider the  gaseous  phase,  the  active  mass  of  a  solid  being  equivalent 
to  its  concentration  in  the  gaseous  phase.  That  is,  every  solid 
is  to  be  looked  upon  as  possessing,  at  a  definite  temperature,  a  defi- 
nite vapor  pressure  which  is  entirely  independent  of  the  amount  of 
solid  present.  Such  substances  as  arsenic,  antimony,  and  cadmium 
are  known  to  have  appreciable  vapor  pressures  at  relatively  low 
temperatures,  and  it  is  quite  reasonable  to  suppose  that  every 
solid  substance  exerts  a  definite  vapor  pressure  at  a  definite  temper- 
ature, even  though  we  have  no  method  sufficiently  refined  to  meas- 
ure such  minute  pressures. 

Since  the  active  mass  of  a  solid  remains  constant  so  long  as 
any  of  it  is  present,  the  application  of  the  law  of  mass  action 
to  certain  heterogeneous  equilibria  is,  in  general,  simpler  than 
its  application  to  homogeneous  systems.  The  truth  of  this  state- 
ment will  be  evident  after  a  few  typical  heterogeneous  systems 
have  been  considered. 

(a)  Dissociation  of  Calcium  Carbonate.     In  the  reaction 
[CaC03]^[CaO]  +  (C02), 

let  TTi  and  ir2  represent  the  pressures  due  to  the  vapor  of  calcium 
carbonate  and  calcium  oxide  respectively,  and  let  p  denote  the 
pressure  of  the  carbon  dioxide.  Applying  the  law  of  mass  action, 
we  obtain 


But  since  TTI  amd  7r2  are  constant  at  any  one  temperature,  the 
equation  becomes 

P  =  K,', 

or,  the  equilibrium  constant  at  any  one  temperature  is  solely 
dependent  upon  the  pressure  of  the  carbon  dioxide  evolved.     The 

*  Loc.  cit. 


330 


THEORETICAL  CHEMISTRY 


accompanying  table  gives  the  values  of  the  pressure  of  carbon 
dioxide  corresponding  to  various  temperatures. 


Temperature, 
Degrees. 

Pressure  in 
Millimeters  of 
Mercury. 

547 

27 

610 

46 

625 

56 

740 

255 

745 

289 

810 

678 

812 

753 

865 

1333 

(b)  Dissociation  of  Ammonium  Hydrosulphide.  When  solid 
ammonium  hydrosulphide  is  heated,  it  is  almost  completely  dis- 
sociated into  ammonia  and  hydrogen  sulphide  as  shown  by  the 
following  equation  :  — 


This  reaction  was  investigated  by  Isambert,*  who  found  that  the 
total  gas  pressure  at  25°.  1  C.  is  equal  to  501  mm.  of  mercury. 
Since  the  partial  pressures  of  the  ammonia  and  hydrogen  sulphide 
are  necessarily  the  same,  each  must  be  approximately  equal  to 
250.5  mm.,  the  relatively  small  pressure  due  to  the  undissociated 
vapor  of  the  ammonium  hydrosulphide  being  neglected.  Let  IT 
be  the  partial  pressure  of  the  vapor  of  ammonium  hydrosulphide, 
and  let  p\  and  p2  be  the  partial  pressures  of  the  ammonia  and 
hydrogen  sulphide.  Applying  the  law  of  mass  action,  we  have 


Pi -Pa  _ 


-IT,-. 


(1) 


Since  TT  is  constant  at  any  one  temperature,  equation  (1)  becomes 

pi'pz  =  Kpf. 
According  to  Dalton's  law  of  partial  pressures,  we  have 

P  =   Pi  +  PZ  +  7T, 

*  Compt.  rend.,  93,  595,  730  (1881). 


HETEROGENEOUS  EQUILIBRIUM 


331 


where  P  is  the  total  pressure.     Neglecting  the  relatively  small 
pressure  w,  we  may  write 

P  =  Pi  +  P2- 

Hence,  since  p\  =  p*, 

P 

2  =  Pi  =  Pz  • 

Substituting  these  values  in  equation  (1),  we  obtain 

(501)' 


P2 

V 

T 


=  62,750 


The  value  of  the  equilibrium  constant  may  be  checked  by  observ- 
ing the  effect  on  the  system  of  the  addition  of  an  excess  of  either 
one  of  the  products  of  the  dissociation.  The  accompanying  table 
gives  the  results  of  a  few  of  Isambert's  experiments. 


Pressure  of 
Ammonia. 

Pressure  of 
Hydrogen 
Sulphide. 

PNH3'PH2S  =  KP- 

208 
138 
417 
453 

294 
458 
146 
143 

61,152 
63,204 
60,882 
64,779 

Mean  62,504 

As  will  be  seen,  the  mean  value  of  the  equilibrium  constant  agrees 
well  with  the  value  found  for  equivalent  amounts  of  the  products 
of  dissociation. 

(c)  Dissociation  of  Ammonium  Carbamate.  The  dissociation  of 
ammonium  carbamate  takes  place  according  to  the  equation 


NH2 

This  dissociation  has  been  investigated  by  Horstmann.*    Applying 
the  law  of  mass  action,  we  have 

Pi2'  P*  =  jr  (2) 


*  Lieb.  Ann.,  187,  48  (1877).| 


332  THEORETICAL  CHEMISTRY 

where  pi  and  p2  are  the  partial  pressures  of  ammonia  and  carbon 
dioxide  respectively,  and  where  IT  is  the  partial  pressure  of  ammon- 
ium carbamate.  Since  IT  is  constant,  equation  (2)  becomes 


If  P  denotes  the  total  gaseous  pressure,  and  TT  is  neglected  as  in 
the  preceding  example,  we  have,  since  three  mols  of  gas  are 
formed 

4P2  P 

Pi2  :    -g-,    and    p2  =  g- 

Substituting  these  values  in  equation  (2),  we  have 

4 


27 


=  Kp'. 


This  equation  has  also  been  tested  by  Isambert  *  by  adding  an 
excess  of  ammonia  or  carbon  dioxide  to  the  dissociating  system. 
He  found  that  the  value  of  the  equilibrium  constant  remains 
practically  constant.  The  addition  of  a  foreign  gas  was  shown 
to  be  without  effect  on  the  dissociation. 

(d)  Dissociation  of  the  Hydrates  of  Copper  Sulphate.  Many  inter- 
esting examples  of  heterogeneous  equilibrium  are  furnished  by  hy- 
drated  salts.  Thus,  if  crystallized  copper  sulphate,  CuSC^.S  H2O, 
is  placed  in  a  desiccator,  it  gradually  loses  water  of  crystallization 
and  ultimately  only  the  anhydrous  salt  remains.  If  the  desiccator 
be  provided  with  a  manometer  and  is  so  arranged  that  the  tem- 
perature can  be  maintained  constant,  it  is  possible  to  observe  the 
changes  in  vapor  pressure  accompanying  the  process  of  dehydra- 
tion. At  the  temperature  of  50°  C.,  the  pressure  over  completely 
hydrated  copper  sulphate  is  found  to  remain  constant  at  47  mm. 
until  the  salt  has  been  deprived  of  two  molecules  of  water,  when 
it  drops  abruptly  to  30  mm.  and  remains  constant  until  two  more 
molecules  of  water  have  been  lost.  It  then  drops  again  to  4.4  mm. 
and  remains  constant  until  dehydration  is  complete. 

The  successive  stages  of  the  dehydration  are  shown  in  the  accom- 
panying diagram,  Fig.  76.  The  constant  pressures  observed  in 
the  dehydration  correspond  to  the  successive  equilibria  involved. 

*  Loc.  cit. 


HETEROGENEOUS  EQUILIBRIUM 


333 


At  50°  C.  the  pentahydrate  and  the  trihydrate  are  in  equilibrium, 
a  pressure  of  47  mm.  being  maintained  so  long  as  any  of  the  penta- 
hydrate is  present.  When  all  of  the  pentahydrate  is  used  up, 
then  the  trihydrate  begins  to  undergo  dehydration  into  the 
monohydrate.  This  is  a  new  equilibrium  and  the  pressure  of  the 


47mm 


30  .mm 


4.5,  mm 


3.H2O 
Composition 

Fig.  76. 


1H2O       OH2O 


aqueous  vapor  necessarily  changes,  and  remains  constant  so  long 
as  any  trihydrate  remains.  The  last  stage  corresponds  to  the 
equilibrium  between  the  monohydrate  and  the  anhydrous  salt. 
The  following  equations  represent  the  three  successive  equilibria:  — 

(1)  CuS04  •  5  H20  ^±  CuS04  •  3  H20  +  2  H20, 

(2)  CuS04  •  3  H20  +±  CuS04  •  H2O  +  2  H20, 

(3)  CuS04  •  H20  <=>  CuS04  +  H2O. 

Applying  the  law^of  mass  action  to  the  first  of  the  above  equi- 
libria, we  have 


in  which  TTI  and  ir2  denote  the  partial  pressures  due  to  the  hydrates 
CuSO4.5  H2O  and  CuSO4.3  H20  respectively,  and  p  denotes  the 


334 


THEORETICAL  CHEMISTRY 


pressure  of  aqueous  vapor.  Since  TTI  and  7r2  are  constant,  the 
above  expression  simplifies  to  the  following 

p2  =  K,'. 

In  a  similar  manner  it  may  be  shown  that  the  pressure  of  aqueous 
vapor  in  the  other  equilibria  must  be  constant.  It  must  be 
clearly  understood  that  the  observed  pressure  is  only  definite 
and  fixed  when  two  hydrates  are  present.  If  the  dehydration 


CuS04 


Temperature 
Fig.  77. 

were  conducted  at  another  temperature  than  50°  C.  the  equilibrium 
pressure  would  be  different.  The  vapor  pressure  curves  of  the 
different  hydrates  are  shown  in  the  temperature-pressure  diagram 
of  Fig.  77. 

Heat  of  Dissociation  of  Solids.  When  the  products  of  the 
dissociation"  of  a  solid  are  gaseous,  it  has  been  pointed  out  by  De 
Forcrand  '*  that  the  ratio  of  the  heat  of  dissociation  of  1  mol  of 

*  Ann.  Chim.  Phys.  [7],  28,  545. 


HETEROGENEOUS  EQUILIBRIUM 


335 


solid  to  the  absolute  temperature  at  which  the  dissociation  pres- 
sure is  equal  to  1  atmosphere,  is  constant.  Or,  denoting  the  heat 
of  dissociation  by  Q  and  the  absolute  temperature  by  T,  De  For- 
crand's  relation  may  be  expressed  thus, 


~  =  constant 


33. 


Nernst  has  shown  that  the  value  of  the  constant  in  this  relation 
is  not  independent  of  the  temperature.  Thus,  the  value  of  the 
ratio  at  100°  C.  is  29.7,  while  at  1000°  C.  it  is  37.7.  Up  to  the 
present  time  no  expression  has  been  derived  in  which  the  variation 
of  the  ratio  with  the  temperature  is  included. 

Distribution  of  a  Solute  between  Two  Immiscible  Solvents. 
When  an  aqueous  solution  of  succinic  acid  is  shaken  with  ether, 
the  acid  distributes  itself  between  the  ether  and  the  water  in  such 
a  way  that  the  ratio  between  the  two  concentrations  is  always 
constant.  It  will  be  seen  that  the  distribution  of  the  succinic 
acid  between  the  two  solvents  is  analogous  to  that  of  a  substance 
between  the  liquid  and  gaseous  phases  (see  page  170),  and  there- 
fore the  laws  governing  the  latter  equilibrium  should  apply  equally 
to  the  former.  Nernst  *  has  shown  that  (a)  //  the  molecular 
weight  of  the  solute  is  the  same  in  both  solvents,  the  ratio  in  which  it 
distributes  itself  between  them  is  constant  at  constant  temperature, 
or  in  other  words,  Henry's  law  is  applicable;  and  (b)  //  there  are 
several  solutes  in  solution  the  distribution  of  each  solute  is  the  same  as 
if  it  were  present  alone.  This  is  clearly  Dalton's  law  of  partial 
pressures.  The  ratio  in  which  the  solute  distributes  itself  between 
the  two  solvents  is  termed  the  coefficient  of  distribution  or  partition. 
The  following  table  gives  the  results  of  three  experiments  on  the 
distribution  of  succinic  acid  between  ether  and  water. 


Concentration 
in  Water. 

Concentration 
in  Ether. 

Distribution 
Coefficient. 

43.4 

7.1 

6.1 

43.8 

7.4 

5.9 

47.4 

7.9 

6.0 

Zeit.  phys.  Chem.,  8,  110  (1891), 


336 


THEORETICAL  CHEMISTRY 


As  will  be  seen  the  distribution  coefficient  is  constant,  show- 
ing that  Henry's  law  applies.  When  the  molecular  weight  of  a 
solute  is  not  the  same  in  both  solvents  the  distribution  coefficient 
is  not  constant,  and  conversely,  if  the  distribution  coefficient  is 
not  constant,  we  infer  that  the  molecular  weights  of  the  solute 
in  the  two  solvents  are  not  identical. 

Let  us  assume  that  a  solute  whose  normal  molecular  weight  is 
A,  when  shaken  with  two  immiscible  solvents  undergoes  polymeri- 
zation in  one  of  them,  its  molecular  weight  being  An.  We  then 
have  the  equilibrium 

An^±nA: 

applying  the  law  of  mass  action,  we  have 


If  the  molecular  weight  in  one  solvent  is  twice  the  molecular 
weight  in  the  other,  then  n  =  2,  and 

—      or     — 7^  =  constant,  since  the  two  ratios  are  not  equal. 

C2  VCa 

Thus  Nernst  found  the  following  concentrations  of  benzoic  acid 
when  it  was  shaken  with  benzene  and  water. 


Cl  (Water). 

c2  (Benzene). 

C2 

V=' 

0.0150 
0.0195 
0.0289 

0.242 
0.412 
0.970 

0.062 
0.048 
0.030 

0.0305 
0.0304 
0.0293 

As  will  be  seen,  the  values  of  the  ratio  Ci/c2  steadily  Decrease, 
while  on  the  other  hand,  the  values  of  the  ratio  Ci/Vc^  remain 
constant.  This  shows,  therefore,  that  benzoic  acid  has  twice  the 
normal  molecular  weight  in  benzene. 

The  Solution  of  a  Solid  in  a  Non-dissociating  Solvent.  When 
a  solid  is  brought  in  contact  with  a  non-dissociating  solvent,  it 
continues  to  dissolve  until  the  solution  becomes  saturated.  A 
condition  of  equilibrium  then  obtains,  the  rates  of  solution  and 


HETEROGENEOUS  EQUILIBRIUM  337 

precipitation  being  the  same.  This  is  plainly  a  case  of  hetero- 
geneous equilibrium.  If  c  is  the  concentration  of  the  dissolved 
substance,  and  IT  is  the  concentration  of  the  undissolved  solid,  then 
according  to  the  law  of  mass  action 

c      K 
-=K*> 

or  since  IT  is  constant, 

c  =  Kcr. 

Variation  of  the  Constant  of  Heterogeneous  Equilibrium  with 
Temperature.     The  reaction  isochore  equation  of  van't  Hoff 

d  (log«g)      _Q_ 
dT          RT2' 

which  has  been  shown  to  connect  the  displacement  of  a  homo- 
geneous equilibrium  with  change  in  temperature,  applies  equally 
well  to  heterogeneous  equilibria.  The  following  examples  will 
serve  to  illustrate  its  application  in  such  cases. 

(a)  Dissociation  of  Ammonium  Hydrosulphide.     In  the  reaction 
representing  the  dissociation  of  ammonium  hydrosulphide, 

[NH4HS]^(NH3)  +  (H2S), 

let  pi  and  p2  be  the  partial  pressures  of  ammonia  and  hydrogen 
sulphide,  and  let  TT  be  the  partial  pressure  of  ammonium  hydro- 
sulphide.  Then  as  has  been  shown  (see  page  331), 


where  P  is  the  total  gaseous  pressure.     From  the  following  data:  — 

Ti  =  273°  +  9°.5,  Pi  =  175  mm.  of  mercury, 

and 

r2  =  273°  +  25°.l,  P2  =  501  mm.  of  mercury, 

we  have,  on  applying  the  reaction  isochore  equation,  and  solving 
forQp, 

4.581  [log  (^i)2  -  log  (lpj]  282.5  X  298.1 
®p  =  282.5  -  298.1 

or  Qp  =  —  22,740  calories. 


338  THEORETICAL  CHEMISTRY 

This  result  agrees  well  with  the  value,  —  22,800  calories,  found 
by  direct  experiment. 

(b)  Solution  of  Succinic  Acid.     The  concentration  of  succinic 
acid  (in  a  saturated  solution)  and  the  temperature,  are  the  factors 
which  determine  the  equilibrium  in  this  case.     In  the  equation 
d  (\ogeKc)        Qc 
dT  RT2' 

Kc  =  c,  where  c  is  the  concentration  of  succinic  acid  in  a  saturated 
solution.  The  following  experimental  data,  due  to  van't  Hoff, 
enables  us  to  calculate  the  heat  of  solution  of  the  acid. 

Ti  =  273°  c  =  2.88  grams  per  100  grams  of  water, 

and 

T2  =  273°  +  8°.5,       c  =  4.22  grams  per  100  grams  of  water. 

Substituting  in  the  reaction  isochore  equation  and  solving  for  Qc, 
we  have 

Q       4.581  (log  4.22  -  log  2.88)  273  X  281.5 

^c  =  273  -  281.5 

or 

Q  =  —  6871  calories. 

The  value  of  the  heat  of  solution  for  1  mol  of  succinic  acid  as 
found  by  direct  experiment  is  —6700  calories. 

The  Phase  Rule.  While  it  is  possible  to  apply  the  law  of 
mass  action  to  certain  heterogeneous  equilibria  there  are  numerous 
cases  where  its  application  is  either  difficult  or  impossible.  To 
deal  with  such  heterogeneous  systems  we  make  use  of  a  general- 
ization discovered  by  J.  Willard  Gibbs,*  late  professor  of  mathe- 
matical physics  in  Yale  University.  This  generalization  was  first 
stated  by  Gibbs  in  1874,  and  is  commonly  known  as  the  phase  rule. 
Before  entering  upon  a  discussion  of  the  phase  rule,  it  will  be 
necessary  to  define  a  few  of  the  terms  employed. 

The  composition  of  a  system  is  determined  by  the  number  of 
independent  variables  or  components  involved.  Thus  in  the 
system  —  ice,  water,  and  vapor  —  there  is  but  a  single  com- 
ponent. In  the  system 

CaC03<=±CaO  +  CO2, 
*  Trans.  Connecticut  Academy,  Vols.  II  and  III,  1875-8. 


HETEROGENEOUS  EQUILIBRIUM  339 

while  there  are  three  constituents  of  the  equilibrium,  only  two  of 
these  need  be  considered  as  components,  for  the  amount  of  any 
one  constituent  is  not  independent  of  the  amounts  of  the  other 
two,  as  the  following  equations  show :  — 

CaO  +  CO2  =  CaC03, 

CaC03  -  CaO  =  CO2, 

CaC03  -  C02  =  CaO. 

In  general,  the  components  are  chosen  from  the  smallest  number 
of  independently-variable  constituents  required  to  express  the 
composition  of  each  phase  entering  into  the  equilibrium,  even 
negative  quantities  of  the  components  being  permissible. 

The  number  of  variable  factors,  —  temperature,  pressure,  and 
concentration, —  of  the  components  which  must  be  arbitrarily  fixed 
in  order  to  define  the  condition  of  the  system,  is  known  as  the 
degree  of  freedom  of  the  system.  For  example,  a  gas  has  two 
degrees  of  freedom  since  two  of  the  variables,  temperature,  pres- 
sure or  volume,  must  be  fixed  in  order  to  define  it;  a  liquid  and  its 
vapor  has  only  one  degree  of  freedom,  since  for  equilibrium  at  a 
certain  temperature,  there  can  be  but  a  single  pressure;  and  in  a 
system  consisting  of  a  substance  in  the  three  states  of  aggregation, 
equilibrium  can  only  exist  at  a  single  temperature  and  pressure. 

Derivation  of  the  Phase  Rule.  The  following  derivation  of 
the  phase  rule  is  due  to  Nernst.  Let  us  assume  a  complete  hetero- 
geneous equilibrium  made  up  of  y  phases  of  n  components,  and  let 
us  fix  our  attention  upon  one  single  phase.  This  phase  will  con- 
tain a  certain  amount  of  each  one  of  the  n  components,  the  con- 
centrations of  which  may  be  designated  by  Ci,  c2,  c3,  .  .  .  cn. 
Since  we  have  assumed  complete  equilibrium  to  exist,  the  slightest 
change  in  concentration,  temperature  or  pressure  will  alter  the 
composition  of  this  phase. 

This  may  be  expressed  by  the  equation 

/  (ci,  C2,  c3,  .  .  .  cn,  p,  T)  =  0, 

where  /  is  any  function  of  the  variables.  Since  any  change  in 
one  phase  implies  a  corresponding  change  in  the  remaining  y  —  1 
phases,  it  follows  that  the  composition  of  all  the  phases  is  a  certain 
determined  function  of  the  same  variables. 


340  THEORETICAL  CHEMISTRY 

The  above  equation  is,  then,  of  the  form  ascribed  to  each  sepa- 
rate phase,  and  since  there  are  y  phases  we  have  y  separate  equa- 
tions. There  are,  however,  n  +  2  variables  in  each  equation,  so 
that  if  y  =  n  +  2,  that  is  if  we  have  two  more  phases  than  com- 
ponents, each  unknown  quantity  has  a  definite  known  value. 
In  this  case  there  is  only  one  value  for  ci,  c*,  c3,  c4,  .  .  .  cn,  p  and 
T  at  which  the  system  can  be  in  equilibrium.  Hence  when  n 
components  are  present  in  n  +  2  phases,  we  have  equilibrium  only 
for  a  certain  temperature,  a  certain  pressure,  and  a  certain  ratio 
of  concentrations  of  the  single  phases.  That  is,  n  +  2  phases  of  n 
substances  can  only  exist  at  a  certain  point  in  a  co-ordinate 
system.  This  point  is  termed  the  transition  point.  If  one  value 
be  altered  then  one  phase  vanishes,  and  there  remain  n  +  1  phases 
of  n  components,  and  the  problem  becomes  indeterminate.  Thus 
it  is  proved  that  n  components  are  necessary  in  order  that  a  system 
containing  n  -j-  1  phases  may  exist  in  complete  equilibrium. 

The- phase  rule  may  be  stated  as  follows:  —  A  system  made  up 
of  n  components  in  n  +  2  phases  can  only  exist  when  pressure, 
temperature  and  concentration  have  definite  fixed  values;  a  system 
of  n  components  in  n  +  1  phases  can  exist  only  so  long  as  one  of  the 
factors  varies;  and  a  system  of  n  components  in  n  phases  can  exist 
only  so  long  as  two  of  the  factors  vary.  If  P  denotes  the  number  of 
phases,  C  the  number  of  components,  and  F  the  number  of  degrees 
of  freedom,  then  the  phase  rule  may  be  conveniently  summarized 
by  the  expression, 

C  -  P  +  2  =  F. 

Equilibrium  in  the  System,  Water,  Ice,  and  Vapor.  In  this 
system  we  may  have  one,  two,  or  three  phases  present,  according 
to  the  conditions.  Under  ordinary  circumstances  of  temper- 
ature and  pressure,  water  and  water  vapor  are  in  equilibrium. 
The  vapor  pressure  curve  of  water  is  represented  by  the  line  OA 
in  the  pressure-temperature  diagram  (Fig.  78).  It  is  only  at 
points  on  this  curve  that  water  and  its  vapor  are  in  equilibrium. 
Thus,  if  the  pressure  be  reduced  below  that  corresponding  to  any 
point  on  OA,  all  of  the  water  will  be  vaporized;  if  on  the  other 
hand,  the  pressure  be  raised  above  the  curve,  all  of  the  vapor  will 


HETEROGENEOUS  EQUILIBRIUM 


341 


ultimately  condense  to  the  liquid  state.  When  the  temperature 
is  reduced  below  0°  C.,  only  ice  and  vapor  are  present,  the  curve 
OC  representing  the  equilibrium  between  these  two  phases.  It  is 
to  be  observed  that  the  curve  OC  is  not  continuous  with  OA.  At 


Solid 


Vapor 


0.0075 

Temperature 
Fig.  78. 

the  point  0,  where  the  two  curves  intersect,  ice,  water,  and  water 
vapor  are  in  equilibrium.  At  this  point  ice  and  water  must  have 
the  same  vapor  pressure,  otherwise  distillation  of  vapor  from  the 
phase  having  the  higher  vapor  pressure  to  that  with  the  lower 
vapor  pressure  would  occur,  and  eventually  the  phase  having  the 
higher  vapor  pressure  would  disappear.  This  result  would  be 
in  contradiction  to  the  experimentally-determined  fact  that 
both  solid  and  liquid  phases  are  in  equilibrium  at  the  point  0. 
The  temperature  at  which  ice  and  water  are  in  equilibrium  with 
their  vapor  under  atmospheric  pressure  is  0°  C.  Since  increase 
of  pressure  lowers  the  freezing-point  of  water,  the  point  0,  repre- 
senting the  equilibrium  between  ice  and  water  under  the  pressure 
of  their  own  vapor,  viz.,  4.57  mm.,  must  be  a  little  above  0°  C. 
The  exact  temperature  corresponding  to  the  point  0  has  been 
found  to  be  O.°0075  C. 


342  THEORETICAL  CHEMISTRY 

The  change  in  the  melting-point  of  ice  due  to  increasing  pressure 
is  represented  by  the  line  OB.  This  line  is  inclined  toward  the 
vertical  axis  because  the  melting  point  of  ice  is  lowered  by  in- 
creased pressure.  The  point  0  is  called  a  triple  point  because 
there,  and  there  only,  three  phases  are  in  equilibrium.  As  is  well 
known,  water  does  not  always  freeze  exactly  at  0°  C.  If  the 
containing  vessel  is  perfectly  clean,  and  care  is  taken  to  exclude 
dust,  it  is  possible  to  supercool  water  several  degrees  below  its 
freezing-point  and  measure  its  vapor  pressure. 

The  dotted  curve  OA',  which  is  a  continuation  of  OA,  represents 
the  vapor  pressure  of  supercooled  water.  It  will  be  noticed  that 
(1)  there  is  no  break  in  the  vapor-pressure  curve  so  long  as  the 
solid  phase  does  not  separate,  and  (2)  the  vapor  pressure  of  super- 
cooled water,  which  is  an  unstable  phase,  is  greater  than  that  of 
ice,  the  stable  phase,  at  that  temperature. 

We  now  proceed  to  apply  the  phase  rule  to  this  system.  In  the 
formula,  C  -  P  +  2  =  F,  C  =  1.  It  is  evident  that  if  P  =  3, 
then  F  =  0;  or  the  system  has  no  degree  of  freedom.  We  have 
seen  that  the  triple  point  0,  represents  such  a  condition.  At  this 
point  ice,  water,  and  water  vapor  are  co-existent,  and  if  either 
one  of  the  variables,  temperature  or  pressure,  is  altered,  one  of 
the  phases  disappears;  in  other  words,  the  system  has  no  degree 
of  freedom.  Such  a  system  is  said  to  be  non-variant.  If  in  the 
above  formula,  P  =  2,  then  F  =  1,  and  the  system  has  one  degree 
of  freedom,  or  is  univariant.  Any  point  on  any  one  of  the  curves 
OA,  OB,  or  OC  represents  a  univariant  system.  Take,  for  exam- 
ple, a  point  on  the  curve  OA.  In  this  case  the  temperature  may 
be  altered  without  altering  the  number  of  phases  in  equilibrium. 
If  the  temperature  is  raised,  a  corresponding  increase  in  vapor 
pressure  follows  and  the  system  will  adjust  itself  to  some  other 
point  on  the  curve  OA.  In  like  manner,  the  pressure  may  be 
altered  without  causing  the  disappearance  of  one  of  the  phases. 
If,  however,  the  temperature  is  maintained  constant,  then  a  change 
in  the  pressure  will  cause  either  condensation  of  water  vapor  or 
vaporization  of  liquid  water.  Under  these  conditions  the  system 
has  only  one  degree  of  freedom.  Again,  if  P  =  1,  then  F  =  2, 
and  the  system  is  bivariant,  or  has  two  degrees  of  freedom.  The 


HETEROGENEOUS  EQUILIBRIUM  343 

areas  included  between  the  curves  in  the  diagram  are  examples 
of  bivariant  systems.  Consider  the  vapor  phase;  the  temperature 
may  be  fixed  at  any  desired  value  within  the  vapor  area  AOC, 
and  the  pressure  may  be  altered  along  a  line  parallel  to  the  vertical 
axis  without  causing  a  change  in  the  number  of  phases,  provided 
the  curves  OA  and  OC  are  not  intersected. 

The  System,  Sulphur  (Rhombic,  Monoclinic),  Liquid  and 
Vapor.  This  system  is  more  complicated  than  the  preceding 
one-component  system,  since  there  are  two  solid  phases  in  addition 
to  the  liquid  and  vapor  phases.  At  ordinary  temperatures,  rhom- 
bic sulphur  is  the  stable  modification.  When  this  is  heated 
rapidly  it  melts  at  115°  C.,  but  if  it  is  maintained  in  the  neighbor- 
hood of  100°  C.  it  gradually  changes  into  monoclinic  sulphur 
which  melts  at  120°  C.  Monoclinic  sulphur  can  be  kept  indefin- 
itely at  100°  C.  without  undergoing  change  into  the  rhombic 
modification,  or  in  other  words  it  is  the  stable  phase  at  this  temper- 
ature. 

It  is  evident,  therefore,  that  there  must  be  a  temperature  above 
which  monoclinic  sulphur  is  the  stable  form  and  below  which 
rhombic  sulphur  is  the  stable  modification.  This  temperature 
at  which  both  rhombic  and  monoclinic  modifications  are  in  equi- 
librium with  each  other  and  with  their  vapor,  is  termed  the 
transition  point.  Its  value  has  been  determined  to  be  95°.6  C. 
The  change  from  one  form  into  the  other  is  relatively  slow,  so 
that  it  is  possible  to  measure  the  vapor  pressure  of  rhombic  sul- 
phur up  to  its  melting-point,  and  that  of  monoclinic  sulphur 
below  its  transition  point.  The  vapor  pressure  of  solid  sulphur, 
although  very  small,  has  been  measured  as  low  as  50°  C. 

The  complete  pressure-temperature  diagram  for  sulphur  is 
shown  in  Fig.  79.  At  the  point  0,  rhombic  and  monoclinic  sul- 
phur are  in  equilibrium  with  sulphur  vapor,  this  being  a  triple 
point  analogous  to  the  point  0  in  Fig.  78.  The  vapor  pressure 
curves  of  rhombic  and  monoclinic  sulphur  are  represented  by  OB 
and  OA  respectively.  The  dotted  curve  OA'  which  is  a  continu- 
ation of  OA  is  the  vapor-pressure  curve  of  monoclinic  sulphur  in 
a  metastable  region.  In  like  manner  OB'  represents  the  vapor- 
pressure  curve  of  rhombic  sulphur  in  the  metastable  condition,  B* 


344 


THEORETICAL  CHEMISTRY 


being  a  metastable  melting  point.  As  in  the  pressure-temper- 
ature diagram  for  water,  the  metastable  phases  have  the  higher 
vapor  pressures.  The  effect  of  increasing  pressure  on  the  transi- 
tion point  0,  is  represented  by  the  line  OC.  This  is  termed  a 


Vapor 


Temperature 
Fig.  79. 

transition  curve,  and,  since  increase  in  pressure  raises  the  transi- 
tion point,  the  line  slopes  away  from  the  vertical  axis.  The 
effect  of  increased  pressure  on  the  melting-point  of  monoclinic 
sulphur  is  shown  by  the  curve  AC. 

This  also  slopes  away  from  the  vertical  axis,  but  the  change  in 
the  melting-point  of  monoclinic  sulphur  produced  by  a  given 
change  in  pressure  being  less  than  the  corresponding  change  in  the 


HETEROGENEOUS  EQUILIBRIUM  345 

transition  point,  the  two  curves,  OC  and  AC,  intersect  at  the 
point  C.  The  point  C  corresponds  to  a  temperature  of  131°  C. 
and  a  pressure  of  400  atmospheres.  The  vapor-pressure  curve 
of  stable  liquid  sulphur  is  represented  by  the  curve  AD.  The 
vapor-pressure  curve  of  the  metastable  liquid  phase  is  represented 
by  the  curve  ABr  which  is  continuous  with  AD.  The  diagram  is 
completed  by  the  curve  B'C  which  represents  the  effect  of  pressure 
on  the  metastable  melting-point  of  rhombic  sulphur.  Mono- 
clinic  sulphur  does  not  exist  above  the  point  C;  hence  when 
liquid  sulphur  is  allowed  to  solidify  at  pressures  exceeding  400  at- 
mospheres, the  rhombic  modification  is  formed,  whereas  under 
ordinary  pressures  the  monoclinic  modification  appears  first. 

The  phase  rule  enables  us  to  state  the  exact  conditions  required 
for  equilibrium  in  this  system  and  to  check  the  results  of  exper- 
iment. Thus,  according  to  the  formula,  C  —  P  +  2  =  F,  since 
C  =  1,  the  system  will  be  non-variant  when  P  =  3.  Since  there 
are  four  phases  involved,  theoretically  any  three  of  these  may 
be  co-existent  and  four  triple  points  are  possible.  The  theoreti- 
cally-possible triple  points  are  as  follows:  — 

(1)  Rhombic  sulphur,  monoclinic  sulphur,  and  vapor  (0); 

(2)  Rhombic  sulphur,  monoclinic  sulphur,  and  liquid  (C); 

(3)  Rhombic  sulphur,  liquid,  and  vapor  (Bf); 

(4)  Monoclinic  sulphur,  liquid  and  vapor  (A). 

In  this  particular  system  all  of  the  four  possible  triple  points  .can 
be  realized  experimentally.  That  this  is  the  case  is  due  to  the 
comparative  slowness  of  the  change  from  rhombic  to  monoclinic 
sulphur  above  the  triple  point.  If  this  change  were  rapid  it  is 
evident  that  all  of  the  theoretically-possible  non-variant  systems 
could  not  be  realized  experimentally. 

As  in  the  case  of  water,  the  curves  in  the  diagram  represent 
univariant  systems  and  the  areas  bivariant  systems.  The  student 
is  advised  to  tabulate  the  univariant  and  bivariant  systems  repre- 
sented in  the  pressure-temperature  diagram  for  sulphur. 

Two-component  Systems.  Turning  now  to  two-component 
systems  we  are  confronted  with  a  more  difficult  problem,  and  one 
which  includes  many  special  cases.  Thus,  we  may  have  cases  of 


346 


THEORETICAL  CHEMISTRY 


anhydrous  salts  and  water,  hydrated  salts  and  water,  volatile 
solutes,  two  liquid  phases,  consolute  liquids,  and  solid  solutions. 
To  enter  upon  a  discussion  of  these  would  not  be  profitable,  since 
they  only  serve  to  give  greater  emphasis  to  the  general  truth  of 
the  phase  rule.  We  shall  select  a  few  typical  two-component 
systems  for  consideration  here. 

(a)  Anhydrous  Salt  and  Water.     In  the  equilibrium  diagram 
of  water  (here  represented  by  dotted  lines,  Fig.  80),  we  desig- 


Temperature 
Fig.  80. 

nate  the  triple  point  by  0.  At  this  point  ice  and  water  have  the 
same  vapor  pressure.  Similarly,  a  solution  at  its  freezing-point 
has  the  same  vapor  pressure  as  the  ice  which  separates.  The 
intersection  of  the  vapor-pressure  curve  for  ice,  OB,  and  the  vapor- 
pressure  curve  of  the  solution  of  the  anhydrous  salt,  0"A",  deter- 
mines a  new  triple  point  0" '.  Since  the  presence  of  the  dissolved 
salt  tends  to  diminish  the  vapor  pressure  of  water,  the  curve  0" A" 
is  situated  below  the  curve  OA,  and  for  the  same  reason  the  triple 
point  0"  is  found  to  the  left  of  0.  If  now  we  keep  an  excess  of 
dissolved  substance  continually  present,  all  of  the  liquid  phases 


HETEROGENEOUS  EQUILIBRIUM  347 

which  are  formed  will  of  necessity  be  saturated  solutions.  When 
these  solutions  finally  freeze  they  will  furnish,  not  pure  ice,  but  a 
mixture  of  ice  and  solid  salt,  known  as  a  cryohydrate.  By  a 
partial  freezing  we  can  therefore  obtain  the  system:  Solid  salt, 
ice,  saturated  solution  and  vapor,  or  in  other  words,  a  system  of 
n  -f-  2  phases  of  which  the  existence  is  only  possible  at  the  freez- 
ing temperature  T'  of  the  saturated  solution,  and  under  the  pres- 
sure p'  corresponding  to  the  vapor  pressure  of  ice  and  the  saturated 
solution.  These  conditions  are  represented  in  the  diagram  by 
the  quadruple  point  O1 '.  If  now  we  pass  from  the  point  0',  increas- 
ing the  temperature  and  pressure  as  prescribed  by  the  curve 
O'A',  the  ice  disappears,  while  the  salt,  the  saturated  solution, 
and  the  vapor  furnish  a  series  of  3-phase  systems.  Again  start- 
ing from  the  point  0'  and  lowering  the  temperature  and  the  pres- 
sure as  indicated  by  the  curve  O'B,  the  liquid  phase  disappears, 
while  the  solid  salt,  ice,  and  vapor  constitute  another  series  of 
3-phase  systems.  This,  of  course,  is  on  the  supposition  that  the 
vapor  pressure  of  the  solid  salt  is  negligible.  Finally,  a  consider- 
able increase  in  pressure  causes  a  slight  lowering  of  the  temper- 
ature corresponding  to  the  quadruple  point,  the  conditions  being 
represented  by  the  curve  O'C". 

All  possible  non-saturated  solutions  of  the  salt  will  be  repre- 
sented by  points  within  the  area,  AOO'A'.  Thus,  let  0"A"  repre- 
sent the  vapor-pressure  curve  of  a  dilute  solution  of  the  salt  in 
water.  The  freezing-point  of  this  solution  is  represented  by  the 
point  0",  while  0"C"  represents  the  variation  of  the  freezing-point 
of  the  solution  with  pressure. 

The  following  table  summarizes  the  possibilities  indicated  by 
the  phase  rule :  — 

,    4  phases;   salt,  ice,  saturated  solution,  vapor  (point,  0')', 
3  phases;  salt,  saturated  solution,  vapor  (curve,  O'A'); 
3  phases;  salt,  ice,  vapor  (curve,  O'B)', 
3  phases;  salt,  ice,  saturated  solution  (curve,,  O'C'); 
2  phases;  salt,  saturated  solution  (area,  C'O'A')', 
2  phases;  salt,  water  vapor  (area,  BO' A')', 
2  phases;  salt,  ice  (area,  BO'C')] 


348 


THEORETICAL  CHEMISTRY 


3  phases;  ice,  non-saturated  solution,  vapor  (curve,  00'); 
2  phases;  non-saturated  solution,  vapor 

1  phase;  non-saturated  solution 

2  phases;  non-saturated  solution,  ice 
1  phase;  non-saturated  solution 


(area, 


(area,  COO'C'). 


As  will  be  seen,  there  is  only  one  non-variant  point  in  the  entire 
diagram,  viz.,  the  point  0'.  In  this  system  there  are  three  degrees 
of  freedom,  since  in  addition  to  temperature  and  pressure  the  con- 
centration of  the  solution  may  also  be  varied. 

The  pressure-temperature  diagram  (Fig.  80)  having  been  dis- 
cussed, we  now  turn  to  the  concentration-temperature  diagram 
for  the  same  system,  Fig.  81.  In  this  diagram  the  abscissae 


Temperature 
Fig.  81. 

represent  temperatures  and  the  ordinates,  concentrations.  For 
convenience,  corresponding  points  in  Figs.  80  and  81  will  be  desig- 
nated by  the  same  letters.  The  equilibrium  between  ice,  water 
and  water  vapor  is  represented  by  the  point  0.  If  now  a  small 


HETEROGENEOUS  EQUILIBRIUM  349 

amount  of  anhydrous  salt  be  added  to  the  water,  the  freezing-point 
will  be  lowered  to  0".  As  the  proportion  of  salt  is  increased  the 
temperature  of  equilibrium  is  lowered  along  the  curve  00" 0'.  A 
point  is  ultimately  reached  at  which  the  solution  becomes  saturated, 
and  on  further  addition  of  salt  it  is  not  dissolved,  but  remains  in 
contact  with  the  ice  and  saturated  solution.  This  is  the  cryo- 
hydric  point,  and  represents  the  lowest  temperature  which  can  be 
obtained  in  this  particular  system.  The  diagram  is  completed 
by  the  solubility  curve  of  the  salt,  O'A'.  Each  point  on  this 
curve  represents  the  concentration  of  the  saturated  solution  at 
all  temperatures,  from  the  critical  temperature  of  the  solution  to 
the  cryohydric  temperature.  The  meaning  of  the  concentration- 
temperature  diagram  may  be  made  clearer  by  a  consideration  of 
the  behavior  of  a  solution  when  gradually  cooled.  Let  a  repre- 
sent a  dilute  solution  of  the  anhydrous  salt.  On  lowering  the 
temperature  along  ab,  no  change  will  occur  until  the  curve  00' 
is  reached;  then  ice  will  begin  to  separate  and  as  the  cooling  is 
continued,  the  composition  of  the  solution  will  change  along  00' 
until  it  reaches  the  cryohydric  point  0'.  Here  both  salt  and  ice 
will  separate,  and  the  solution  will  solidify  completely  at  the 
temperature  corresponding  to  the  point  0'.  In  like  manner,  if 
we  start  with  a  concentrated  solution  represented  by  the  point  c 
and  cool  along  cd  no  change  will  take  place  until  the  curve  O'A' 
is  reached;  then  solid  salt  will  separate  and  the  composition  of 
the  solution  will  alter  along  O'A'  until  the  temperature  is  reduced 
to  that  corresponding  to  the  cryohydric  point,  when  the  whole 
solution  will  solidify  as  in  the  previous  case.  This  phenomenon 
was  first  systematically  investigated  by  Guthrie  *  who  concluded 
that  such  mixtures  of  constant  composition  and  definite  melting- 
point  are  chemical  compounds,  and,  therefore,  he  proposed  to  call 
them  cryohydrates.  It  has  since  been  shown  that  cryohydrates 
are  not  definite  chemical  compounds.  Among  the  various  reasons 
which  have  been  advanced  to  prove  the  incorrectness  of  Guthrie's 
views,  the  following  are  the  most  cogent: — (1)  the  physical 
properties  of  a  cryohydrate  are  the  mean  of  the  corresponding 
properties  of  the  constituents,  this  being  rarely  true  of  chemical 
*  Phil.  Mag,  [4],  49,  1  (1875);  [5],  i,  49  and  2,  211  (1876). 


350 


THEORETICAL  CHEMISTRY 


compounds;  (2)  the  lack  of  homogeneity  of  a  cryohydrate  can 
be  detected  under  the  microscope;  and  (3)  the  constituents  are 
seldom  present  in  simple  molecular  proportions. 

Applying  the  phase  rule  to  the  above  two-component  system, 
it  is  evident  that  there  is  but  one  non-variant  system:  this  is 
represented  by  the  point  0' '.  When  three  phases  are  co-existent 
the  system  is  univa  riant,  when  only  two  phases  are  present  the 
system  is  bivariant,  and  finally,  when  only  one  phase  is  present 
the  system  acquires  three  degrees  of  freedom  or  is  trivariant. 
It  is  evident  that  a  system  having  three  degrees  of  freedom  cannot 
be  completely  represented  by  a  diagram  in  a  single  plane.  It  is 
possible,  however,  to  construct  a  three-dimensional  model  which 
will  represent  the  equilibrium  very  satisfactorily.  Such  a  model  is 


Temperature 


Fig.    82. 

I.  Unsaturated  Solution. 

II.  Salt  and  Saturated  Solution. 

III.  Ice  and  Unsaturated  Solution. 

IV.  Ice  and  Cryohydrate. 
V.  Salt  and  Cryohydrate. 

shown  in  Fig.  82,  the  lettering  being  made  to  correspond  with 
that  of  the  two  diagrams,  Figs.  81  and  82,  from  which  it  is  derived. 


HETEROGENEOUS  EQUILIBRIUM 


351 


(b)  Hydrated  Salt  and  Water.  An  interesting  example  is  fur- 
nished by  the  system  —  ferric  chloride  and  water.  This  system 
has  been  very  carefully  investigated  by  Roozeboom.*  The  con- 
centration-temperature diagram,  plotted  from  Roozeboom's  data, 


Temperature  - 
Fig.  83. 

is  given  in  Fig.  83.  The  freezing-point  of  pure  water  is  repre- 
sented by  A,  and  the  lowering  of  the  freezing-point  produced  by 
the  addition  of  ferric  chloride  is  indicated  by  the  curve  AB.  At 
the  cryohydric  temperature,  -  55°  C.,  ice,  Fe2Cle  •  12  H20,  sat- 
urated solution,  and  vapor  are  in  equilibrium,  and  the  system  is 
non-variant.  On  adding  more  ferric  chloride,  the  ice  phase  dis- 
appears, and  the  univariant  system,  Fe2Cl6  •  12  H20,  saturated 
solution,  and  vapor  results.  The  equilibrium  is  represented  by  the 
curve  BC  which  may  be  regarded  as  the  solubility  curve  of  the 
dodecahydrate.  On  continuing  the  addition  of  ferric  chloride, 
the  temperature  continues  to  rise  until  the  point  C  is  reached. 
Here  the  composition  of  the  solution  is  identical  with  that  of  the 
dodecahydrate,  and,  therefore,  the  temperature  corresponding  to 
this  point,  37°  C.,  may  be  looked  upon  as  the  melting-point  of 
Fe2Cl6  •  12  H20.  Further  addition  of  ferric  chloride  will  naturally 
*  Zeit.  phys.  Chem.,  4,  31  (1889);  10,  477  (1892). 


352  THEORETICAL  CHEMISTRY 

lower  the  melting  point  and  the  equilibrium  will  alter  along  the 
curve  CD.  It  is  thus  possible  to  have  two  saturated  solutions, 
one  of  which  contains  more  water  and  the  other  less,  than  the 
hydrate  which  is  in  equilibrium  with  the  solution.  These  solu- 
tions are  both  stable  throughout  and  are  nowhere  supersaturated. 
Roozeboom  was  the  first  investigator  to  discover  a  saturated 
solution  containing  less  water  than  the  solid  hydrate  with  which 
it  is  in  equilibrium.  This  discovery  led  him  to  define  supersatu- 
ration  as  follows:  —  "A  solution  is  supersaturated  with  respect 
to  a  solid  phase  at  a  given  temperature  if  its  composition  is  between 
that  of  the  solid  phase  and  the  saturated  solution. "  At  the  point 
D  the  curve  reaches  another  minimum  which  is  analogous  to  the 
point  B,  except  that  the  heptahydrate,  Fe2Cl6  •  7  H20,  takes  the 
place  of  ice.  Here  we  have  equilibrium  between  the  dodecahydrate, 
the  heptahydrate,  saturated  solution,  and  vapor,  and  the  system 
is  non-variant.  On  further  addition  of  ferric  chloride  another 
maximum  is  reached  at  E,  corresponding  to  the  melting-point  of 
the  heptahydrate.  In  a  similar  manner,  two  other  maxima  at 
greater  concentrations  of  ferric  chloride  reveal  the  existence  of  the 
hydrates,  Fe2Cl6  •  5  H2O,  and  Fe2Cl6  •  4  H20. 

At  the  three  remaining  quadruple  points  the  following  phases  are 
in  equilibrium : — At  F,  Fe2Cl6'7  H20,  Fe2Cl6-5  H20,  saturated  solu- 
tion and  vapor;  at  H,  Fe2Cle  •  5  H20,  Fe2Cl6  •  4  H2O,  saturated 
solution  and  vapor;  and  at  K,  Fe2Cle  •  4  H20,  Fe2Cle,  saturated 
solution  and  vapor.  The  solubility  of  the  anhydrous  salt  is 
represented  by  the  curve  KL.  Metastable  solubility  and  melting- 
point  curves  are  represented  by  dotted  lines. 

The  student  should  apply  the  phase  rule  to  this  system.  If  a 
fairly  dilute  solution  of  ferric  chloride  is  evaporated  at  31°  C.,  the 
water  gradually  disappears  and  a  residue  of  the  dodecahydrate 
remains.  This  residue  then  liquefies  and  again  dries  down,  the 
composition  of  the  residue  corresponding  to  the  heptahydrate: 
on  further  standing  the  phenomenon  is  repeated,  the  final  and 
permanent  residue  having  a  composition  corresponding  to  the 
pentahydrate.  The  dotted  line  ab  shows  the  isothermal  along 
which  the  composition  varies.  It  would  have  been  a  difficult 
matter  to  explain  the  alternations  of  moisture  and  dryness  ob- 


HETEROGENEOUS  EQUILIBRIUM  353 

served  in  this  experiment  without  the  concentration-temperature 
diagram. 

Alloys.  Among  the  most  interesting  two-component  systems 
known  are  those  involving  mixtures  of  metals,  or  alloys.  These 
have  been  made  the  subject  of  systematic  investigations  by  num- 
erous experimenters  among  whom  may  be  mentioned  Roberts- 
Austen,  Charpy,  Roozeboom,  and  Heycock  and  Neville.  We  have 
space  to  consider  only  two  comparatively-simple  cases. 

(a)  Alloys  of  Silver  and  Copper.  The  conditions  of  equilibrium 
in  this  binary  system  have  been  studied  by  Heycock  and  Neville.* 
The  two  components,  silver  and  copper,  are  not  miscible  in  the 
solid  state  and  do  not  combine  chemically.  To  determine  the 
curves  of  equilibrium,  mixtures  of  the  two  metals  in  varying  pro- 
portions were  fused  and  then  allowed  to  cool  slowly,  the  rate  of 
cooling  being  observed  with  a  thermocouple,  one  junction  of  which 
was  maintained  at  constant  temperature,  while  the  other  junction 
was  placed  in  the  mixture  of  molten  metals.  The  terminals  of 
the  thermocouple  were  connected  to  a  sensitive  galvanometer 
graduated  to  read  directly  in  degrees,  and  the  rate  of  cooling 
was  followed  by  the  movement  of  the  needle  of  the  galvanometer. 
As  the  mixture  cooled,  two  " breaks"  were  observed;  the  first  of 
these  varied  with  the  composition  of  the  mixture,  while  the  second 
remained  practically  constant  at  777°  C.  When  the  temperatures 
corresponding  to  the  first  break  are  plotted  as  ordinates  against 
the  composition  of  the  mixture  as  abscissae,  the  diagram  shown  in 
Fig.  84  is  obtained. 

The  point  A  represents  the  freezing-point  of  pure  silver,  B  that 
of  pure  copper,  the  curve  AO  represents  the  effect  of  the  gradual 
addition  of  copper  upon  the  freezing-point  of  silver,  and  BO  the 
effect  of  silver  on  the  freezing-point  of  copper.  The  intersection 
of  the  two  curves  at  0  corresponds  to  an  alloy  containing  40  atomic 
per  cent  of  copper.  This  lowest  melting  mixture  is  known  as 
the  eutectic  (ev  =  well,  and  TTTJKUV  =  melt)  mixture.  At  0  the 
system  is  non-variant,  silver,  copper,  solution  and  vapor  being  in 
equilibrium.  The  solid  which  separates  at  0,  having  a  more 
uniform  texture  than  that  of  all  other  mixtures  of  the  two  com- 
*  Phil.  Trans.,  189,  25  (1897). 


354 


THEORETICAL  CHEMISTRY 


ponents,  is  known  as  the  eutectic  alloy.  When  the  composition 
of  a  mixture  of  two  metals  corresponds  to  that  of  the  eutectic 
alloy,  the  two  metals  crystallize  simultaneously  in  minute  separate 


Solution 


Aff+Solution 


Cu+Solution 


AK+ Eutectic 


Cu+Eutectlc 


1081.5 


Ag 


40  at.  per  cent 

Concentration 

Fig.  84. 


Cu 


crystals.  When  examined  under  the  microscope  the  solid  eutectic 
alloy  will  be  seen  to  be  a  conglomerate  of  very  small  crystals, 
whereas  all  of  the  other  alloys  of  the  same  metals  will  be  found  to 
contain  large  crystals  of  either  one  or  the  other  component  em- 
bedded in  the  conglomerate.  While  the  composition  of  the  eutec- 
tic alloy  in  the  above  system  is  found  to  correspond  very  closely 
to  the  formula  Ag3Cu2,  yet  the  nature  of  the  equilibrium  curves 
proves  it  to  be  nothing  more  than  a  mechanical  mixture  of  the 
two  metals.  The  meaning  of  the  diagram  will  be  clearer  from  a 
careful  consideration  of  the  phenomena  accompanying  the  cooling 
of  a  mixture  of  the  molten  metals. 

Take  for  example,  a  fused  mixture  relatively  rich  in  silver.     As 
the  temperature  falls,  a  point  will  ultimately  be  reached  at  which 


HETEROGENEOUS  EQUILIBRIUM 


355 


pure  silver  begins  to  separate,  and  since  the  temperature  remains 
constant  during  the  solidification,  a  break  occurs  in  the  cooling 
curve.  This  first  break  corresponds  to  a  point  on  the  curve  AO. 
As  silver  continues  to  separate,  the  composition  of  the  mixtures 
changes  along  AO,  until  when  0  is  reached,  the  mixture  is  satu- 
rated with  respect  to  copper,  and  both  metals  separate  as  a  con- 
glomerate having  the  same  composition  as  the  fused  mixture. 
The  separation  of  the  eutectic  alloy  causes  the  second  break  in 
the  cooling  curve,  the  temperature  remaining  constant  until  the  en- 
tire mass  has  solidified.  It  will  be  noticed  that  this  system  is  the 
exact  analogue  of  the  system — anhydrous  salt  and  water;  the  eutec- 
tic point  and  the  cryohydric  point  representing  identical  conditions. 
(b)  Alloys  of  Gold  and  Aluminium.  This  system  has  been 
studied  by  Roberts- Austen.*  The  equilibrium  curves  in  the 
concentration  temperature  diagram,  Fig.  85,  reveal  the  existence 


I 


Composition  • 
Fig.   85. 

*  Phil.  Trans.  A.,  194,  201  (1900). 


356  THEORETICAL  CHEMISTRY 

of  definite  compounds,  AusAl2,  Au2Al,  and  AuAl2,  corresponding 
to  the  points  D,  E,  and  H  respectively.  The  discontinuities  at  B 
and  G  suggest  the  possibility  of  two  other  compounds,  viz.,  Au4Al 
and  AuAl.  The  diagram  shows  that  the  following  substances 
will  crystallize  in  succession  from  the  molten  alloy,  these  being 
the  different  solids  with  which  the  liquid  mixture  is  saturated  in 
its  successive  stages  of  equilibrium :  - 

Curve  AD,  pure  gold  at  A ; 

Curve  BC,  Au4Al,  nearly  pure  at  B; 

Curve  CD,  Au5Al2  or  Au8Al3,  nearly  pure  at  D; 

Curve  DEF,  Au2Al,  pure  at  E\ 

Curve  FG,  AuAl,  maximum  undetermined ; 

Curve  GHI,  AuAl2,  pure  at  H', 

Curve  7J,  Al,  pure  at  J. 

The  points  C,  F,  and  /  represent  non-variant  systems,  the  melt- 
ing points  of  the  respective  eutectic  alloys  being  527°,  569°,  and 
647°.  This  system  in  many  respects  resembles  the  system  —  ferric 
chloride  and  water. 

Three-component  Systems.  When  three  components  are  pres- 
ent, the  equilibria  become  much  more  complicated.  Applying 
the  formula,  C  —  P  +  2  =  F,  we  find  that  it  is  necessary  to 
have  five  phases  co-existent  for  a  non-variant  system,  four  for  a 
univariant,  three  for  a  bivariant,  and  two  for  a  trivariant.  The 
most  satisfactory  method  of  representing  equilibria  in  three-com- 
ponent systems  is  that  in  which  use  is  made  of  the  triangular 
diagram.  The  three  corners  of  an  equilateral  triangle  are  taken 
to  represent  the  pure  components,  and  the  composition  of  any 
mixture,  expressed  in  atomic  percentages,  is  represented  by  the 
position  of  the  center  of  mass  of  the  three  components  within 
the  triangle. 

IJor  example,  in  the  system,  —  potassium  nitrate,  sodium  nitrate, 
and  lead  nitrate,  carefully  investigated  by  Guthrie,*  the  three 
components  are  placed  at  the  corners  of  the  triangle  shown  in 

*  Phil.  Mag.,  5,  17,  472  (1884). 


HETEROGENEOUS  EQUILIBRIUM  357 

Fig.  86.     The  melting-point  of  pure  potassium  nitrate  is  340° 
and  that  of  pure  sodium  nitrate  is  305°.     The  melting-point  of 


Fig.  86. 

pure  lead  nitrate  cannot  be  determined  since  the  salt  decomposes 
before  its  melting-point  is  reached.  The  eutectic  mixtures  of 
the  three  pairs  of  salts  are  represented  by  the  points  D,  E,  and  F 
respectively.  In  like  manner  0  represents  the  melting  point  of 
the  non-variant  system,  —  potassium  nitrate,  sodium  nitrate,  lead 
nitrate,  fused  mixture  of  the  three  salts,  and  vapor.  In  order 
to  represent  temperature,  use  is  frequently  made  of  a  triangular 
prism  in  which  the  altitude  is  taken  as  the  temperature  axis,  the 
resulting  surface  within  the  prism  representing  the  variation  of 
the  equilibrium  with  temperature.* 

PROBLEMS. 

1.  The  vapor  pressure  of  solid  NH4HS  at  25°.  1  is  50.1  cm.  Assuming 
that  the  vapor  is  practically  completely  dissociated  into  NH3  and  H2S, 
calculate  the  total  pressure  at  equilibrium  when  solid  NH4HS  is  allowed 

*  For  a  complete  treatment  of  three-component  systems  as  well  as  for  a 
clear  presentation  of  the  phase  rule,  the  student  should  consult  "  The  Phase 
RuJe  and  Its  Applications,"  by  Alexander  Findlay. 


358  THEORETICAL  CHEMISTRY 

to  dissociate  at  25°.  1  in  a  vessel  containing  ammonia  at  a  pressure  of 
32  cm.  Ans.  59.5  cm. 

2.  In  the  partition  of  acetic  acid  between  CCU  and  water,   the  con- 
centration of  the  acetic  acid  in  the  CC14  layer  was  c  gram-molecules  per 
liter  and  in  the  corresponding  water  layer  w  gram-molecules  per  liter. 

c  0.292  0.363  0.725  1.07  1.41 

w          4.87  5.42  7.98  9.69  1.07 

Acetic  acid  has  its  normal  molecular  weight  in  aqueous  solutions.  From 
these  figures  show  that,  at  these  concentrations,  the  acetic  acid  in  the 
carbon  tetrachloride  solution  exists  as  double  molecules. 

3.  Acetic  acid  distributes  itself  between  water  and  benzene  in  such  a 
manner  that  in  a  definite  volume  of  water  there  are  0.245  and  0.314  gram 
of  the  acid,  while  in  an  equal  volume  of  benzene  there  are  0.043  and 
0.071  gram.    What  is  the  molecular  weight  of  acetic  acid  in  benzene, 
assuming  it  to  be  normal  in  water?  Ans.    121.3. 

4.  The  salt  Na2HP04.12  H20  has  a  vapor  pressure  at  15°  of  8.84  mm., 
and  at  17°. 3  of  10.53  mm.     Calculate  the  heat  of  vaporization,  i.e.,  the 
thermal  change  during  the  loss  of  1  mol  of  water  of  crystallization  by 
evaporation.  Ans.    —12,651  cal. 

5.  The  solubility  of  boric  acid  in  water  is  38.45  grams  per  liter  at  13°, 
and  49.09  grams  per  liter  at  20°.    Calculate  the  heat  of  solution  of  boric 
acid  per  mol.  Ans.  —  5,822  cal. 

6.  Plot  the  pressure-temperature  diagram  for  calcium  carbonate  from 
the  table  given  on  p.  280,  and  apply  the  phase  rule. 

7.  Is  it  possible  to  decide  by  the  phase  rule  whether  the  eutectic  alloy 
is  a  mixture  or  a  compound? 


CHAPTER   XVII. 
CHEMICAL  KINETICS. 

Velocity  of  Reaction.  In  the  two  preceding  chapters  we  have 
considered  the  equilibrium  which  is  established  when  the  speeds 
of  the  direct  and  reverse  reactions  have  become  equal.  We  now 
proceed  to  consider  the  velocity  of  individual  reactions.  By  far 
the  greater  number  of  the  reactions  between  inorganic  substances 
proceed  with  such  rapidity  that  it  is  impossible  to  measure  their 
velocities.  Thus,  when  an  acid  is  neutralized  by  a  base,  the  indi- 
cator changes  color  almost  instantly.  There  are  a  few  well- 
known  reactions  which  are  exceptions  to  this  rule;  among  these 
may  be  mentioned  the  oxidation  of  sulphur  dioxide  and  the  de- 
composition of  hydrogen  peroxide.  Both  of  these  reactions  are 
well  adapted  to  kinetic  experiments.  In  organic  chemistry,  on 
the  other  hand,  slow  reactions  are  the  rule  rather  than  the  excep- 
tion. Thus  the  reaction  between  an  alcohol  and  an  acid  forming 
an  ester  and  water,  proceeds  very  slowly  under  ordinary  condi- 
tions and  the  progress  of  the  reaction  may  be  easily  followed. 
By  means  of  the  law  of  mass  action  it  is  possible  to  derive  equations 
expressing  the  velocity  of  a  reaction  at  any  moment  in  terms  of 
the  concentrations  of  the  reacting  substances  present  at  that  time. 

Let  the  equation 


represent  a  reversible  reaction  and  let  a,  6,  c,  and  d  be  the  respec- 
tive initial  concentrations  of  the  reacting  substances  A\,  A2,  A\, 
and  A2'.  The  velocity  of  the  direct  reaction  will  then  be 

AT 

g  =  k  (a  -  x)  (b  -  x),  (D 

where  k  is  the  velocity  constant,  and  dx  is  the  infinitely  small 
increase  in  the  amount  of  x  during  the  infinitely  small  interval 

359 


360  THEORETICAL  CHEMISTRY 

of  time  dt.  Similarly  the  velocity  of  the  reverse  reaction  will 
be 

dx' 

^•  =  *i  (c  +  x)  (d  +  x).  (2) 

It  is  evident  that  the  substances  on  the  right-hand  side  of  the  equa- 
tion will  exert  an  ever-increasing  influence  upon  the  velocity  of 
the  direct  reaction,  which  must  accordingly  decrease.  .When, 
however,  the  velocities  of  the  direct  and  reverse  reactions  become 
equal,  equilibrium  will  be  established,  and  the  ratio  of  the  amounts 
of  the  reacting  substances  on  the  two  sides  of  the  equation  will 
remain  constant.  The  total  velocity  due  to  these  opposing  reac- 
tions will  be 

^j  =  ^-~  =  k(a-x)(b-x)-kt(c  +  x)(d  +  x)     (3) 

j~y 

and  at  equilibrium,  when  -jr  =  0, 

k(a-x)(b-x)  =  k,  (c  +  x)  (d  +  x), 
or 

(c  +  ap  (d  +  x)  _kjf  m 

(a  -  x)  (b  -  x)  "  /d  " 

This  equation  has  been  thoroughly  tested  in  the  two  preceding 
chapters.  Thus,  in  the  reaction 

C2H5OH  +  CH3COOH<=>  CH3COOC2H5  +  H2O, 

Kc  has  been  shown  to  have  the  value,  2.84,  at  ordinary  temper- 
atures. The  velocity  constants  of  the  direct  and  reverse  reactions 
have  also  been  determined,  the  values  being,  k  =  0.000238  and 
ki  =  0.000815.  When  these  values  are  substituted  in  the  equa- 

k 

tion,   T-  =  Kc,  we  obtain  Kc  =  2.92,  a  value  which  agrees  well 
KI 

with  that  found  by  direct  experiment.  The  application  of  equa- 
tion (3)  is  much  simplified  by  the  fact  that  most  reactions  proceed 
nearly  to  completion  in  one  direction,  so  that  the  term  k\  (c  +  x) 
(d  +  #)  will  be  so  small  that  it  may  be  neglected.  We  then  have 

2     =  k  (a  -  *)  (6  -  x),  (5) 


CHEMICAL  KINETICS 


361 


an  equation  expressing  the  velocity  of  the  direct  reaction  in  terms 
of  the  concentrations  of  the  reacting  substances. 

Unimolecular  Reactions.  The  simplest  type  of  chemical 
reaction  is  that  in  which  only  one  substance  undergoes  change 
and  in  which  the  velocity  of  the  reverse  reaction  is  negligible.  The 
decomposition  of  hydrogen  peroxide  is  an  example  of  such  a  reac- 
tion. In  the  presence  of  a  catalyst,  such  as  certain  unorganized 
ferments  or  collodial  platinum,  hydrogen  peroxide  decomposes 
as  represented  by  the  equation, 


This  reaction  is  usually  allowed  to  take  place  in  dilute  aqueous 
solution  so  that  there  is  no  appreciable  alteration  in  the  amount 
of  solvent  throughout  the  entire  course  of  the  reaction.  Further- 
more, the  activity  of  the  catalyst  remains  constant  so  that  the 
course  of  the  reaction  is  wholly  determined  by  the  concentration 
of  the  hydrogen  peroxide.  A  very  satisfactory  catalyst  is  haemase, 
an  enzyme  derived  from  blood.  The  concentration  of  hydrogen 
peroxide  present  at  any  time  during  the  reaction  can  be  deter- 
mined very  simply  by  removing  a  definite  portion  of  the  reaction 
mixture,  adding  an  excess  of  sulphuric  acid  to  destroy  the  activity 
of  the  hsemase,  and  then  titrating  with  a  standard  solution  of 
potassium  permanganate. 

The  following  table  gives  the  results  of  such  an  experiment:  — 


t  (minutes). 

a—'x, 
cc.  KMnO4. 

z. 
cc.  KMnO4. 

k 

o 

46   1 

o 

5 

37.1 

9.0 

0.0435 

10 

29.8 

16.3 

0.0438 

20 

19.6 

26.5 

0.0429 

30 

12.3 

33.8 

0.0440 

50 

5.0 

41.1 

0.0444 

Mean   0.0437 

The  second  column  of  the  table  gives  the  number  of  cubic 
centimeters  of  the  potassium  permanganate  solution  required  to 
oxidize  25  cc.  of  the  reaction  mixture  when  the  time  intervals 


362  THEORETICAL  CHEMISTRY 

recorded  in  the  first  column  have  elapsed  after  the  introduction 
of  the  catalyst.  Since  the  numbers  in  the  second  column  repre- 
sent the  actual  concentration  of  hydrogen  peroxide  present  at 
the  end  of  the  successive  intervals  of  time,  it  is  evident  that  the 
difference  between  these  numbers  and  46.1  cc.  —  the  initial 
concentration  of  hydrogen  peroxide  —  will  give  the  amounts  of 
peroxide  decomposed  in  those  intervals.  These  numbers  are 
recorded  in  the  third  column  of  the  table.  It  will  be  seen  that 
as  the  concentration  of  the  hydrogen  peroxide  decreases  the  rate 
of  the  reaction  diminishes.  Thus,  hi  the  first  interval  of  10  min- 
utes, an  amount  of  hydrogen  peroxide  corresponding  to  46.1  — 
29.8  =  16.3  cc.  of  potassium  permanganate  is  decomposed,  while 
in  the  second  interval  of  10  minutes,  the  amount  of  hydrogen 
peroxide  decomposed  is  equivalent  to  29.8  —  19.6  =  10.2  cc.  of 
potassium  permanganate.  Since  only  a  single  substance  is  under- 
going change,  equation  (5)  simplifies  to  the  following  form:- 

dX  7      /  N 

¥  =  k  (a  -  x). 

* 

It  is  impossible  to  apply  the  equation  in  this  form,  since  in  order 
to  obtain  accurate  titrations,  dt  must  be  taken  fairly  large  and 
during  this  interval  of  time  a  —  x  would  have  diminished.  Approx- 
imate values  of  k  may  be  obtained  by  taking  the  average  value 
of  a  —  x  during  the  interval  dt  within  which  an  amount  dx  of  hydro- 
gen peroxide  is  being  decomposed.  For  example,  let  us  take  the 
interval  between  5  and  10  minutes;  dx  =  16.3  —  9.0  =  7.3  cc., 
dt  =  5  min.,  and  the  average  value  of  a  —  x  is 

37.1  +  29.8 


33.45  cc. 


Substituting  in  the  equation 

we  have 
and 


=  k  X  33.45, 

0 


k  =  0.0436. 


CHEMICAL  KINETICS  363 

Similarly  taking  the  next  interval  between  10  and  20  minutes; 
dx  =  26.5  —  16.3  =  10.2  cc.,  dt  =  10  minutes,  and  the  average 

29  8  - 1    19  6 
value  of  a  —  x  is  -  -  =  24.7  cc.     Substituting   in  the 

2i 

equation  as  before,  we  obtain 

and 

k  =  0.0413. 

As  will  be  seen  these  two  values  of  k  are  not  in  good  agreement, 
although  the  first  value  of  k  agrees  closely  with  the  mean  value 
of  k  given  in  the  fourth  column  of  the  table. 
In  order  to  apply  the  equation 

dx  ^  t  •. 
37  =  k  (a  —  x) 
at 

it  must  be  integrated.* 

The  integration  of  this  equation  may  be  performed  as  follows:  — 

dx      j  f          , 
=  *(«-*), 


therefore 

dx 


=  kdt 


a  —  x 
Integrating,  we  have 

/j  /» 

--    /  k  dt  =  constant  =  C, 
a  —  x      J 

therefore 

—  loge  (a  —  x)  —  kt  =  C. 

In  order  to  determine  C,  the  constant  of  integration,  we  make 
use  of  the  experimental  fact  that  when  t  =  0,  x  =  0.  Substitut- 
ing these  values,  we  have 

—  loge  a  =  C.    , 
Consequently 

loge  a  —  loge  (a  —  x)  =  kt, 
or 

1  a 


*  The  student  who  is  unfamiliar  with  the  Calculus  must  take  the  result 
of  this  calculation  for  granted. 


364  THEORETICAL  CHEMISTRY 

Passing  to  Briggsian  logarithms,  we  obtain 

_  I0g     a      =  0.4343  k. 
t        a  —  x 

By  substituting  in  this  equation  the  corresponding  values  of  a, 
a  —  x,  and  t  from  the  preceding  table,  the  values  of  k  given  in  the 
fourth  column  of  the  table  are  obtained. 
The  equation 

dx 

may  also  be  thrown  into  an  exponential  form,  as  follows:  - 
Since 

1,  a 

we  may  write, 


or 

a  —  x  =  ae~kt, 
and 

x  =  a  (1  -  e~kt). 

In  this  equation  k  may  be  regarded  as  the  fraction  of  the  total 
amount  of  substance  decomposing  in  the  unit  of  time,  provided 
this  unit  is  so  small  that  the  quantity  at  the  end  of  the  time  unit 
is  only  slightly  different  from  that  at  the  beginning.  The  time 
required  for  one-half  of  the  substance  to  change,  is  known  as  the 
period  of  half-change,  T,  and  may  be  calculated  from  k  by  means 
of  the  equation 

log  2  =  0.4343  kT, 
therefore 

•  T  =  0.6943  i, 
k 

or 

|  =  1.443  T. 

Reactions  in  which  only  one  mol  of  a  single  substance  undergoes 
change  are  known  as  unimolecular  reactions,  or  reactions  of  the 


CHEMICAL  KINETICS  365 

first  order.  In  a  unimolecular  reaction,  the  velocity  constant  fc, 
is  independent  of  the  units  in  which  concentration  is  expressed. 
If,  in  the  integrated  equation 


t  becomes  infinite,  then  x  =  a.  In  other  words,  for  finite  values 
of  t,  x  must  always  remain  less  than  a  and  the  reaction  will  never 
proceed  to  completion. 

Another  unimolecular  reaction  which  has  been  thoroughly 
investigated,  is  the  hydrolysis  of  cane  sugar.  When  cane  sugar 
is  dissolved  in  water  containing  a  small  amount  of  free  acid  it  is 
slowly  transformed  into  d-glucose  and  d-fructose.  The  velocity 
of  the  reaction  is  very  small  and  is  dependent  upon  the  strength 
of  the  acid  added.  The  progress  of  the  reaction  may  be  very  easily 
followed  by  means  of  the  polarimeter.  Cane  sugar  itself  is  dex- 
tro-rotatory, while  d-fructose  rotates  the  plane  of  polarization 
more  strongly  to  the  left  than  d-glucose  rotates  it  to  the  right. 
Therefore,  as  the  hydrolysis  proceeds,  the  angle  of  rotation  to  the 
right  steadily  diminishes  until,  when  the  reaction  is  complete,  the 
plane  of  polarization  will  be  found  to  be  rotated  to  the  left.  On 
this  account  the  hydrolysis  of  cane  sugar  is  commonly  termed 
inversion  and  the  molecular  mixture  of  d-fructose  and  d-glucose 
constituting  the  product  of  the  reaction  is  called  invert  sugar. 
Let  ao  denote  the  initial  angle  of  rotation,  at  the  time  t  =  0, 
due  to  a  mols  of  cane  sugar,  let  aQ'  denote  the  angle  of  rotation 
when  inversion  is  complete  and  let  a  be  the  angle  of  rotation  at 
any  time  t;  then  since  rotation  of  the  plane  of  polarization  is  pro- 
portional to  the  concentration  x,  the  amount  of  cane  sugar  in- 
verted, will  be 

OQ  —  a 
x  =  a — — — 7- 

CiQ-\-  OQ 

In  the  equation 

C12H22On  •  +  H20  <=±  C6H1206  +C6HJ206, 

representing  the  inversion  of  cane  sugar,  the  velocity  of  the 
reaction  will  be,  according  to  the  law  of  mass  action,  proportional 
to  the  molecular  concentrations  of  the  cane  sugar  and  the  water. 


366 


THEORETICAL  CHEMISTRY 


Since  the  reaction  takes  place  in  the  presence  of  such  a  large  excess 
of  water,  its  effect  may  be  considered  to  be  constant.  The 
velocity  of  the  reaction  is  then  proportional  to  the  active  mass 
of  the  sugar  alone,  or  in  other  words  the  reaction  is  unimolecular. 
In  the  differential  equation  expressing  the  velocity  of  a  unimolec- 
ular reaction, 

dx 


we  have 


,       1 

K  =  - 

t 


a  -  x 


and  since  a  and  x  are  measured  in  terms  of  angles  of  rotation  of 
the  plane  of  polarization,  .we  have 


7          , 

fc  =  7loge-       —  ,• 
t          a  +  a0 

The  following  table  gives  the  results  obtained  with  a  20  per  cent 
solution  of  cane  sugar  in  the  presence  of  0.5  molar  solution  of  lactic 
acid  at  25°  C. 


t  (minutes). 

a 

/; 

0 

34°.  5 

1,435 

31°.  1 

'0^2348' 

4,315 

25°.  0 

0.2359 

7,070 

20°.  16 

0.2343 

11,360 

13°.  98 

0.2310 

14,170 

10°.  01 

0.2301 

16,935 

7°.  57 

0.2316 

19,815 

5°.  08 

0.2991 

29,925 

-  1°.65 

0.2330 

Inf. 

-10°.  77 

Bimolecular  Reactions.  When  two  substances  react  and  the 
concentration  of  each  changes,  the  reaction  is  bimolecular  or  of 
the  second  order.  Let  a  and  b  represent  the  initial  molar  con- 
centrations of  the  two  reacting  substances  and  let  x  denote  the 
amount  transformed  in  the  interval  of  time  t;  then  the  velocity 
of  the  reaction  will  be  expressed  by  the  equation 
dx 


dt 


=  k  (a  —  x)  (b  —  x). 


CHEMICAL  KINETICS  367 

The  simplest  case  is  that  in  which  the  two  substances  are  present 
in  equivalent  amounts.  Under  these  conditions  the  velocity 
equation  becomes 


This  equation  may  be  integrated  as  follows:  —  * 

Mi  =  ^W'  •  ' 

k  r\dt=  r*  dx 

J  i,          J*,   (a  -  x)* 
therefore 


or 


fe  -  ti)     (a  -  xi)  (a  -  Xz)  ' 

If  time  be  reckoned  from  the  beginning  of  the  reaction,  then  Xi  =  0 
and  t  =  0,  and  we  have 


. 

iv    —  '     ,  t  \ 

t     a  (a  —  x) 

If  the  reacting  substances  are  not  present  in  equivalent  amounts 
then  the  velocity  equation  becomes 

~  =  k  (a  -  x)  (b  -  x). 

Assuming  that  time  is  measured  from  the  beginning  of  the  reaction, 
the  integration  of  this  equation  may  be  performed  as  follows:  — 


_ 

—  x)  (b  —  x) 
Decomposing  into  partial  fractions, 


_ 

a  —  b    Jo  b  —  x 


*  The  student  who  is  unfamiliar  with  the  Calculus  must  take  the  results 
of  these  calculations  for  granted. 


368  THEORETICAL  CHEMISTRY 

therefore 


14          1     fi       a-xf 

kt  = r    logc  r i 

a  -  b  \_    3  6  -  x Jo 


or 

u  =      1  6  (o  -  re) 

Or  passing  to  Briggsian  logarithms, 
0.4343  k 


t  (a  —  6)     *a(b  —  x) 

The  value  of  fc  in  a  bimolecular  reaction  is  not  independent  of 
the  units  in  which  the  concentration  is  expressed,  as  is  the  case 
with  a  unimolecular  reaction.  Suppose  that  a  unit  I/nth  of 
that  originally  selected  is  used  to  express  concentration,  then  the 
value  of  k  in  the  equation 


t  a(a  —  x) ' 
becomes 

, ,  _  1  nx  1 


t    na  •  n  (a  —  x)      t    na  (a  —  x) 

Thus,  the  value  of  k  varies  inversely  as  the  numbers  expressing 
the  concentrations. 

As  an  illustration  of  a  bimolecular  reaction  we  may  take  the 
hydrolysis  of  an  ester  by  an  alkali.     The  reaction 

CH3COOC2H5  +  NaOH  <=±  CH3COONa  +  C2H5OH, 

has  been  studied  by  Warder,*  Reicher,f  Arrhenius,  t  Ostwald  § 
and  others.  Arrhenius  employed  in  his  experiments  0.02  molar 
solutions  of  ester  and  alkali.  These  solutions  were  placed  in 
separate  flasks  and  warmed  to  25°  C.  in  a  thermostat  maintained 
at  that  temperature;  equal  volumes  were  then  mixed,  and  at 
frequent  intervals  a  portion  of  the  reaction  mixture  was  removed 

*  Berichte,  14,  1361  (1881). 

t  Lieb.  Ann.,  228,  257  (1885). 

J  Zeit.  phys.  Chem.,  i,  110  (1887). 

§  Jour,  prakt.  Chem.,  35,  112  (1887). 


CHEMICAL  KINETICS 


369 


and  titrated   rapidly  with   standard  acid.     The  accompanying 
table  contains  some  of  the  results  obtained :  — 


t  (minutes). 

a—x 

1 

0 

8.04 

4 

5.30 

0.0160 

6 

4.58 

0.0156 

8 

3.91 

0.0164 

10 

3.51 

0.0160 

12 

3.12 

0.0162 

Mean  0.0160 

The  numbers  in  the  second  column  of  the  table  represent 
concentrations  of  sodium  hydroxide  and  of  ethyl  acetate,  expressed 
in  terms  of  the  number  of  cubic  centimeters  of  standard  acid 
required  to  neutralize  10  cc.  of  the  reaction  mixture.  Owing  to 
the  high  velocity  of  the  reaction  it  is  difficult  to  avoid  large  experi- 
mental errors,  nevertheless  the  values  of  k  given  in  the  third  column 
of  the  table  will  be  observed  to  differ  very  slightly  from  the  mean 
value. 

Reicher  investigated  the  same  reaction  when  the  reacting  sub- 
stances were  not  present  in  equivalent  proportions.  In  this  case, 
the  progress  of  the  reaction  was  followed  by  titrating  definite 
portions  of  the  reaction  mixture  from  time  to  time,  the  excess  of 
sodium  hydroxide  being  determined  by  titrating  a  portion  of 
the  mixture  at  the  expiration  of  twenty-four  hours,  when  the 
ester  was  completely  hydrolyzed.  His  results  are  given  in  the 
following  table :  — 


t  (minutes). 

a—  x 
(alkali). 

b-x 
(ester). 

k 

o 

61  95 

47  03 

4.89 
11.36 
29.18 
Inf 

50.59 
42.40 
29.35 
14  92 

35.67 
27.48 
14.43 
0 

0.00093 
0.00094 
0.00092 

370  THEORETICAL  CHEMISTRY 

Reicher  also  studied  the  effect  of  different  bases  upon  the  ve- 
locity of  the  reaction.  He  found  for  strong  bases  approximately 
equal  values  of  k,  but  for  weak  bases  the  values  were  irregular 
and  smaller  than  those  obtained  with  the  more  completely  ionized 
bases.  Arrhenius  pointed  out  that  the  hydrolyzing  power  of  a 
base  is  proportional  to  the  number  of  hydroxyl  ions  which  it 
yields.  Writing  the  equation  for  the  above  hydrolysis  in  terms  of 
ions,  we  have 

CH3COOC2H5  +  Na*  +  OH'  <=±  CH3COO'  +  Na*  +  C2H5OH. 

It  is  evident  from  this  equation  that  all  bases  furnishing  the  same 
number  of  hydroxyl  ions  should  give  identical  values  of  k.  We 
may,  therefore,  modify  the  fundamental  differential  equation  as 

rfollows :  — 

Af 

^  =  k'a  (a  -x)(b-  x), 

where  a  is  the  degree  of  ionization  of  the  base. 

Trimolecular  Reactions.  When  equivalent  quantities  of  three 
substances  react,  the  reaction  is  trimolecular  or  of  the  third  order. 
If  the  initial  molar  concentrations  of  the  reacting  substances  are 
denoted  by  a,  b,  and  c,  and  if  x  denotes  the  proportion  of  each 
which  is  transformed  in  the  interval  of  time  t,  the  velocity  of  the 
reaction  will  be  represented  by  the  differential  equation 

dx      •,  , 

-^  =  k  (a  -  x)  (b  -  x}(c  -  x). 

If  the  substances  are  present  in  equivalent  amounts,  the  equation 
becomes 

-^  =  k  (a  —  z)3, 

an  expression  which  is  much  less  difficult  to  integrate. 

The  integration  of  this  equation  may  be  performed  as  follows.* 

S*    A  /"»_  1 


*  The  student  who  is  unfamiliar  with  the  Calculus  must  take  the  results 
of  these  calculations  for  granted. 


CHEMICAL  KINETICS  371 

therefore 

*_., 


hence 


*[-J-  T 

2  |_(a- *)*_]„ ' 

=  if       1         -II 
2l(a-x)*      a"J 


,       1     x  (2  a  -  x) 

~     * 


t     2a2(a-a02 

When  the  reacting  substances  are  not  taken  in  equivalent  amounts, 
the  integration  of  the  velocity  equation  may  be  performed  as 
follows  :  — 

~  =  k  (a  -  x)  (b  -  x)  (c  -  x), 

therefore 

,   ,.  dx 


(a  —  x)  (b  —  x)  (c  —  x) 
Decomposing  into  partial  fractions, 


B  C 


(a  —  x)  (b  —  x)  (c  —  x)      a  —  x      b  —  x      c  —  x 
Multiplying  through  by  (a  —  z),  we  obtain 

1  ,  (     B  C 

(a  —  x)  <  7— 
' 


\  \ 

(b  -  x)  (c  -  x)  '  (b  -  x      c  -  x 

Let  x  =  a,  then 

A  ___  i 

(a  -  6)  (c  -  a) 

Similarly,  multiplying  by  (b  —  x)  and  (c  —  x),  and  then  placing 
x  =  b,  and  x  =  c,  we  have 

B=    ~  (a-b)  (b-c)' 
and 

C=  1 

(b-c)(c-  a)' 

Then  we  ottain  by  substitution 

r*  _  dx_  _  =  __  i         r*  dx 

Jo   (a  -  x)  (b  -  x)  (c-  x)         i  (a  -  b)  (c-  a)J0  a-  x' 
__  1  r*    dx  ___  1  Cx    dx 

(a  -  b)  (b  -  c)  Jo   b  -  x      (b-c)  (c-a)J0  c  -  x 


372 


THEORETICAL  CHEMISTRY 


Therefore, 


or 


/c  =  4 


-      «>  ~  c> 


-  [ 


&  - 


(a  -  6)6  -  c) 


~  a> 


(a  -  6)  (b  -  c)  (c  -  a) 

In  a  trimolecular  reaction,  k  is  inversely  proportional  to  the  square 
of  the  original  concentration. 

A  typical  trimolecular  reaction  is  that  between  ferric  and 
stannous  chlorides.  This  reaction,  represented  by  the  following 
equation 

2  FeCl3  +  SnCl2  ?=*  2  FeCl2  +  SnCl4, 

has  been  investigated  by  A.  A.  Noyes.*  Dilute  solutions  of  the 
reacting  substances  were  mixed  at  constant  temperature,  and 
definite  portions  of  the  reaction  mixture  were  removed  at  meas- 
ured intervals  of  time  and  titrated  for  ferrous  iron.  Before 
titrating  with  a  standard  solution  of  potassium  permanganate  it 
was  necessary  to  decompose  the  stannous  chloride  present  with 
mercuric  chloride.  The  following  table  gives  the  results  obtained 
with  0.025  molar  solutions  of  ferric  chloride  and  stannous  chloride. 


t  (minutes). 

a—  x 

X 

it 

2.5 

0.02149 

0.00351 

113 

3 

0.02112 

0.00388 

107 

6 

0.01837 

0.00663 

114 

11 

0.01554 

0.00946 

116 

15 

0.01394 

0.01106 

118 

18 

0.01313 

0.01187 

117 

30 

0.01060 

0.01440 

122 

60 

0.00784 

0.01716 

122 

Mean  116 

Noyes  also  found  that  the  velocity  of  the  reaction  is  accelerated 
more  by  an  excess  of  ferric  chloride  than  by  an  equal  excess  of 
stannous  chloride. 

*  Zeit.  phys.  Chem.,  16,  546  (1895). 


CHEMICAL  KINETICS  373 

Reactions  of  Higher  Orders.  Reactions  of  the  fourth,  fifth 
and  eighth  orders  have  recently  been  investigated,  but  examples 
of  reactions  of  orders  higher  than  the  third  are  extremely  rare. 
This  fact  is  at  first  sight  surprising  since  the  equations  of  many 
chemical  reactions  involve  a  large  number  of  molecules,  and  we 
would  naturally  expect  the  order  of  such  reactions  to  be  corre- 
spondingly high.  For  example,  the  reaction  represented  by  the 
equation, 


involves  six  molecules  of  the  substances  initially  present  and, 
therefore,  we  should  infer  it  to  be  a  reaction  of  the  sixth  order. 

Kinetic  experiments  by  van  der  Stadt  have  shown  it  to  be  a 
bimolecular  reaction,  the  velocity  of  reaction  being  proportional 
to  the  concentrations  of  the  phosphine  and  the  oxygen.  On 
allowing  the  gases  to  mix  slowly  by  diffusion,  it  was  discovered 
that  the  reaction  actually  takes  place  in  several  successive  stages, 
the  first  stage  being  represented  by  the  equation  of  the  bimolec- 
ular  reaction  PH3  +  02  =  HPO2  +  H2. 

The  subsequent  changes,  involving  the  oxidation  of  the  products 
of  this  reaction,  take  place  with  great  rapidity.  It  is  highly 
probable  that  the  equations  which  are  ordinarily  employed  to 
represent  chemical  reactions  really  represent  only  the  initial  and 
final  stages  of  a  series  of  relatively  simple  reactions.  Larmor* 
has  shown  that  when  chemical  reactions  are  considered  from  the 
molecular  standpoint,  the  bimolecular  reaction  is  the  most  prob- 
able. He  says,  "  Imagine  a  substance,  say  gaseous  for  simplicity, 
formed  by  the  immediate  spontaneous  combination  of  three  gas- 
eous components  A,  B,  and  C.  When  these  gases  are  mixed,  the 
chances  are  very  remote  of  the  occurrence  of  the  simultaneous 
triple  encounter  of  an  A,  a  B,  and  a  C,  which  would  be  necessary 
to  the  immediate  formation  of  an  ABC',  whereas  if  ever  formed, 
it  would  be  liable  to  the  normal  chance  of  dissociating  by  collisions; 
it  would  thus  be  practically  non-existent  in  the  statistical  sense. 
But  if  an  intermediate  combination  AB  could  exist,  very  tran- 
siently, though  long  enough  to  cover  a  considerable  fraction  of  the 
*  Proc.  Manchester  Phil.  Soc.,  1908. 


374  THEORETICAL  CHEMISTRY 

mean  free  path  of  the  molecules,  this  will  readily  be  formed  by 
ordinary  binary  encounters  of  A  and  B,  and  another  binary 
encounter  of  AB  with  C  will  now  form  the  triple  compound  ABC 
in  quantity. " 

Determination  of  the  Order  of  a  Reaction.  It  has  been 
shown  in  the  foregoing  pages  that  the  time  required  to  complete 
a  certain  fraction  of  a  reaction  is  dependent  upon  the  order  of 
the  reaction  in  the  following  manner :  — 

(1)  In  a  unimolecular  reaction  the  value  of  k  is  independent  of 
the  initial  concentration; 

(2)  In  a  bimolecular  reaction  the  value  of  k  is  inversely  pro- 
portional to  the  initial  concentration; 

(3)  In  a  trimolecular  reaction  the  value  of  k  is  inversely  pro- 
portional to  the  square  of  the  initial  concentration. 

Hence,  in  general,  in  a  reaction  of  the  nth.  order,  the  value  of 
k  is  inversely  proportional  to  the  (n  —  1)  power  of  the  initial  con- 
centration. If  the  value  of  k  is  determined  with  definite  concen- 
trations of  the  reacting  substances,  and  then  with  multiples  of 
those  concentrations,  the  order  of  the  reaction  can  be  determined 
according  to  the  above  rules  by  observing  the  manner  in  which  k 
varies  with  the  concentration. 

The  order  of  a  reaction  may  also  be  readily  determined  by  means 
of  a  graphic  method.  Thus,  to  determine  the  order  of  a  reaction 
we  ascertain  by  actual  trial  which  one  of  the  following  expressions, 
in  which  C  denotes  concentration,  will  give  a  straight  line  when 
plotted  against  times  as  abscissae :  — 

(1)  log  C  —  reaction  unimolecular; 

(2)  1/C  -—reaction  bimolecular; 

(3)  1/C2  —  reaction  trimolecular; 

(4)  l/Cn  —  reaction  n  -j-  1  molecular. 

Complex  Reaction  Velocities.  Thus  far  we  have  considered 
the  velocity  of  reactions  which  are  practically  complete.  There 
are  numerous  cases,  however,  in  which  the  course  of  the  reaction 
is  complicated  by  such  disturbing  factors  as  (1)  counter  reactions, 
(2)  side  reactions,  and  (3)  consecutive  reactions.  These  disturb- 
ing causes  will  now  be  considered. 


CHEMICAL   KINETICS  375 

(1)  Counter  Reactions.     In  the  chemical  change  represented  by 
the  equation 

CH3COOH  +  C2H5OH  <=±  CH3COOC2H5  +  H2O, 

the  speed  of  the  direct  reaction  steadily  diminishes  owing  to  the 
ever-increasing  effect  of  the  reverse  or  counter  reaction.  Ulti- 
mately, when  two-thirds  of  the  acid  and  alcohol  are  decomposed, 
the  velocities  of  the  two  reactions  become  equal  and  a  condition 
of  equilibrium  results.  Starting  with  1  mol  of  acid  and  1  mol 
of  alcohol,  and  letting  x  represent  the  amount  of  ester  formed, 
we  have 


When  equilibrium  is  attained, 

K  =  —  =  4- 

By  observing  the  change  for  any  time  t,  we  have 

7/       3  .        2  -  x 
k~k  =  T7log«o—Tf-;- 


Having  the  values  of  k/k'  and  k  —  k',  the  velocity  constant  k 
of  the  direct  reaction  can  be  determined.  The  value  of  k  so 
obtained  has  been  shown  by  Knoblauch  *  to  vary  in  those  reac- 
tions where  the  concentration  of  the  hydrogen  ion  changes. 

(2)  Side  Reactions.  When  the  same  substances  are  capable  of 
reacting  in  more  than  one  way  with  the  formation  of  different 
products,  the  several  reactions  proceed  side  by  side.  Thus, 
benzene  and  chlorine  may  react  in  two  ways  as  shown  by  the 
equations, 

(1)  C6H6  +  C12  =  C6H5C1  +  HC1, 
and 

(2)  C6H6  +  3  C18  =  C6H6C16. 

It  is  generally  possible  to  regulate  the  conditions  under  which 
the  substances  react  so  as  to  promote  one  reaction  and  retard  the 

other. 

*  Zeit.  phys.  Chem.,  22,  268  (1897). 


376  THEORETICAL  CHEMISTRY 

(3)  Consecutive  Reactions.  By  consecutive  reactions  we  under- 
stand those  reactions  in  which  the  products  of  a  certain  initial 
chemical  change  react,  either  with  each  other  or  with  the  original 
substances  to  form  new  substances.  Attention  has  already  been 
called  to  the  fact  that  many  of  our  common  chemical  equations 
really  represent  the  summation  of  a  number  of  consecutive  reac- 
tions. If  the  system  A  is  transformed  into  the  system  C  through 
an  intermediate  system  B,  then  we  shall  have  the  two  reactions 

(1)  A-+B, 
and 

(2)  5->C. 

If  reaction  (1)  should  have  a  very  much  greater  velocity  than 
reaction  (2),  then  the  measured  velocity  of  the  change  from  A  to 
C  will  be  practically  the  same  as  that  of  the  slower  reaction. 
This  fact  has  been  illustrated  by  means  of  the  following  analogy, 
due  to  James  Walker:  —  *  "The  time  occupied  by  the  transmission 
of  a  telegraphic  message  depends  both  on  the  rate  of  transmission 
along  the  conducting  wire,  and  on  the  rate  of  progress  of  the 
messenger  who  delivers  the  telegram;  but  it  is  obviously  this 
last,  slower  rate  that  is  of  really  practical  importance  in  determin- 
ing the  time  of  transmission."  The  saponification  of  ethyl 
succinate  may  be  taken  as  an  illustration  of  consecutive  reactions. 
This  reaction  proceeds  in  two  stages  as  follows:  — 


(1)     C2H4  +  NaOH<=±C2H4  +C2H5OH, 


.COONa 

(2)     C2H4  +  NaOH  <r»  C2H4  /  +  C2H5OH. 

N^OONa  XJOONa 

In  this  case  the  product  of  the  first  reaction  reacts  with  one  of  the 
original  substances. 

Velocity  of  Heterogeneous  Reactions.     It  has  been  shown  that 
when  a  solid,  such  as  calcium  carbonate,  is  dissolved  in  an  acid, 

*  Proc.  Roy.  Soc.,  Edinburgh,  22  (1898). 


CHEMICAL  KINETICS  377 

the  rate  of  solution  is  dependent  upon  the  surface  of  contact 
between  the  solid  and  'liquid  phases,  and  also  upon  the  strength 
of  the  acid.  If  the  surface  is  large  so  that  it  undergoes  relatively 
little  change  during  the  reaction,  it  may  be  considered  as  constant. 
If  S  represents  the  area  of  the  surface  exposed  and  x  denotes  the 
amount  of  solid  dissolved  in  the  time  t,  the  velocity  of  the  reaction 
will  be  represented  by  the  differential  equation 


Integrating  this  equation,  we  have 


t     "  a  -  x 

This  formula  has  been  tested  by  Boguski  *  for  the  reaction 
CaC03  +  2  HC1  =  CaCl2  +  CO2  +  H20, 

and  is  found  to  give  constant  values  of  k.  Furthermore,  Noyes  and 
Whitney  f  have  shown  that  the  rate  of  solution  of  a  solid  in  a  liquid 
at  any  instant,  is  proportional  to  the  difference  between  the  con- 
centration of  the  saturated  solution  and  the  concentration  of  the 
solution  at  the  time  of  the  experiment. 

Velocity  of  Reaction  and  Temperature.  It  is  a  well-estab- 
lished fact  that  the  velocity  of  a  chemical  reaction  is  accelerated 
by  rise  of  temperature.  Thus,  the  rate  of  inversion  of  cane  sugar 
is  increased  about  five  times  for  a  rise  in  temperature  of  30°.  It 
has  been  shown  as  the  result  of  a  large  number  of  observations  on 
a  variety  of  chemical  reactions,  that  in  general  the  velocity  of  a 
reaction  is  doubled  or  trebled  for  an  increase  in  temperature  of 
10°.  It  is  of  interest  to  note  that  the  rate  of  development  of 
various  organisms,  such  as  yeast  cells,  the  rate  of  growth  of  the 
eggs  of  certain  fishes,  and  the  rate  of  germination  of  certain 
varieties  of  seeds  is  either  doubled  or  trebled  for  a  rise  in  temper- 
ature of  10°.  Up  to  the  present  time  no  wholly  satisfactory  form- 
ula, connecting  the  rate  of  reaction  with  the  temperature,  has  been 
derived,  although  several  purely-empirical  expressions  have  been 

*  Berichte,  9,  1646  (1876). 

t  Zeit.  phys.  Chem.,  23,  689  (1897). 


378 


THEORETICAL  CHEMISTRY 


suggested.  Of  these  formulas  the  most  widely  applicable  is  that. 
proposed  by  van't  Hoff  and  verified  by  Arrhenius.  If  kQ  and  k\ 
represent  the  velocity  constants  at  the  respective  temperatures 
To  and  Ti,  then 


A  (Px- 


where  e  is  the  base  of  the  Naperian  system  of  logarithms  and  A  is 
a  constant.  The  following  table  gives  the  calculated  and  observed 
values  of  k  at  various  temperatures  for  the  reaction 


when  T  =  273°  +  25°,  k  =  0.000227  and  A  =  11,700. 


T, 
Degrees 

K  (observed), 

k  (calculated). 

273  +  39 

0.00141 

0.00133 

273  +  50.1 

0.00520 

0.00480 

273  +  64.5 

0.0228 

0.0227 

273  +  74.7 

0.062 

0.0623 

273  +  80 

0.100 

0.105 

In  this  case  the  agreement  between  the  observed  and  calculated 
values  is  all  that  could  be  desired. 

Influence  of  the  Solvent  on  the  Velocity  of  Reaction.  The 
velocity  of  a  chemical  reaction  varies  greatly  with  the  nature  of 
the  medium  in  which  it  takes  place.  This  subject  has  been 
studied  by  Menschutkin  *  who  has  collected  much  valuable  data, 
as  the  result  of  a  large  number  of  experiments,  on  the  velocity  of 
the  reaction  between  ethyl  iodide  and  triethylamine,  as  represented 
by  the  equation 

C2H5I  +  (C2H6)3N  =  (C2H6)4NI. 

This  reaction  was  allowed  to  take  place  in  a  large  number  of 

different  solvents  and  the  velocity  at  100°  was  measured.     A  few 

*  Zeit.  phys.  Chem.,  6,  41  (1890). 


CHEMICAL  KINETICS 


379 


of  Menschutkin's  results  are  given  in  the  accompanying  table,  in 
which  k  denotes  the  velocity  constant :  — 


Medium  . 

ft 

Medium. 

k 

Hexane  

0  00018 

Ethyl  alcohol 

0  0366 

Ethylether  

0  .  000757 

Methyl  alcohol 

0  0516 

Benzene  

0.00584 

Acetone 

0  0608 

These  figures  show  that  the  velocity  of  the  reaction  is  greatly 
modified  by  the  nature  of  the  medium  in  which  it  takes  place,  the 
velocity  in  hexane  being  less  than  one  three-hundredth  of  that  in 
acetone.  It  is  of  interest  to  note  that  there  is  an  approximate 
parallelism  between  the  values  of  k,  and  the  values  of  the  dielec- 
tric constant  of  the  different  media. 

Catalysis.  It  is  a  familiar  fact  that  the  velocity  of  reaction 
is  frequently  greatly  accelerated  by  the  presence  of  a  foreign  sub- 
stance which  apparently  does  not  participate  in  the  reaction,  and 
which  remains  unchanged  when  the  reaction  is  complete.  For 
example,  cane  sugar  is  inverted  very  slowly  by  pure  water  alone, 
but  when  a  trace  of  acid  is  added  the  reaction  is  greatly  acceler- 
ated. A  substance  which  is  capable  of  exerting  such  an  acceler- 
ating action  is  termed  a  catalyst,  and  the  process  is  known  as 
catalysis.  In  addition  to  the  fact  that  a  relatively-small  amount 
of  a  catalyst  is  capable  of  effecting  the  transformation  of  large 
amounts  of  material,  there  are  two  other  important  character- 
istics of  catalytic  action  which  should  be  mentioned:  viz.,  (a)  a 
catalyst  does  not  initiate  a  reaction  but  simply  promotes  it;  and 
(6)  the  equilibrium  is  not  disturbed  by  the  presence  of  a  catalyst, 
since  the  velocities  of  the  direct  and  reverse  reactions  are  each 
altered  to  the  same  extent.  As  the  result  of  a  series  of  experi- 
ments, Ostwald  concludes  that  the  catalytic  effect  of  acids  in 
hastening  the  inversion  of  cane  sugar  is  directly  proportional  to 
the  concentration  of  the  hydrogen  ion,  and,  in  general,  is  inde- 
pendent of  the  nature  of  the  anion.  Similarly,  the  catalytic  action 
of  bases  may  be  attributed  to  the  hydroxyl  ion,  the  effect  being 
proportional  to  the  concentration  of  this  ion.  In  fact  we  may 


380  THEORETICAL  CHEMISTRY 

formulate  the  following  fundamental  law  of  catalysis :  —  The 
degree  of  catalytic  action  is  directly  proportional  to  the  concentration 
of  the  catalytic  agent.  Almost  every  chemical  reaction  can  be 
accelerated  by  the  addition  of  an  appropriate  catalyst.  A  few 
typical  reactions  which  are  accelerated  catalytically  are  here 
given,  together  with  the  catalyst  employed :  — 

Catalyst  —  hydrogen  ion, 

CH3COOC2H5  +  H20  =  CH3COOH  +  C2H6OH, 

Catalyst  —  hydroxyl  ion, 

2CH3.CO-CH3  =  CH3.CO-CH2C(CH3)2OH, 

Catalyst  —  finely  divided  platinum, 
2  SO2  +  02  =  2  SO3, 
2  CH3OH  +  02  =  2  H-COH  +  2  H20, 
2  H2  +  O2  =  2  H20, 

Catalyst  —  water  vapor, 

2  CO  +  02  =  2  C02, 
NH4C1  =  NH3  +  HC1, 

Catalyst  —  copper  sulphate, 


(Deacon  Process) 

Catalyst  —  mercury  salts, 

COOH+2H20  +  4C02 
-COOH 

(First  step  in  the  synthesis  of  indigo) 

Catalyst  —  colloidal  platinum, 

2  H202  =  2  H20  +  O2, 

Catalyst  —  enzymes, 

C6Hi206  =  2  C2H5OH  +  2  CO2, 

(zymase) 

C3H7COOH  +  C2H3OH  =  C3H7COOC2H5  +  H2O. 

(lipase) 


CHEMICAL  KINETICS  381 

It  will  be  seen  that  catalysis  is  of  great  importance  in  connection 
with  many  industrial  processes  as  well  as  in  the  field  of  pure 
chemistry.  The  majority  of  the  reactions  occurring  within 
living  organisms  are  accelerated  catalytically  by  unorganized 
ferments  or  enzymes.  Thus,  before  the  process  of  digestion  can 
proceed,  starch  must  be  changed  into  sugar.  This  transformation 
is  accelerated  by  an  enzyme  called  ptyalin  occurring  in  the  saliva, 
and  by  other  enzymes  found  in  the  pancreatic  juice.  The  digestion 
of  albumen  is  hastened  by  the  enzymes,  pepsin  and  trypsin.  As 
a  rule  each  enzyme  acts  catalytically  on  just  one  reaction,  or  in 
other  words  the  catalytic  action  of  enzymes  is  specific.  Enzymes 
are  very  sensitive  to  traces  of  certain  toxic  substances  such  as 
hydrocyanic  acid,  iodine,  and  mercuric  chloride. 

An  interesting  series  of  experiments  by  Bredig  *  on  the  catalytic 
action  of  colloidal  metals,  established  the  fact  that  these  substan- 
ces resemble  the  enzymes  very  closely  in  their  behavior.  Thus, 
they  are  "poisoned"  by  the  same  substances  which  inhibit  the 
activity  of  the  enzymes,  and  they  show  the  same  tendency  to 
recover  when  the  amount  of  the  poison  does  not  exceed  a  certain 
limiting  value.  Because  of  this  close  similarity,  Bredig  called  the 
colloidal  metals  inorganic  ferments. 

It  sometimes  happens  that  one  of  the  products  of  a  chemical 
reaction  functions  as  a  catalyst  to  the  reaction.  Thus,  when 
metallic  copper  is  dissolved  in  nitric  acid,  the  reaction  proceeds 
slowly  at  first  and  then,  after  a  short  interval,  the  speed  of  the 
reaction  is  greatly  augmented.  The  acceleration  is  due  to  the 
catalytic  action  of  the  nitric  oxide  evolved.  This  phenomenon 
is  known  as  autocatalysis.  In  reactions  where  autocatalysis 
occurs,  the  velocity  increases  with  the  time  until  a  certain  maximum 
value  is  reached,  after  which  the  velocity  steadily  diminishes.  In 
ordinary  reactions  the  initial  velocity  is  the  greatest. 

It  sometimes  happens  that  the  speed  of  a  reaction  is  retarded 
by  the  presence  of  a  trace  of  some  foreign  substance.  Thus, 
Bigelow  f  has  shown  that  the  rate  of  oxidation  of  sodium  sulphite 
is  retarded  by  the  presence  in  the  solution  of  only  one  one-hundred- 

*  Zeit.  phys.  Chem.,  31,  258  (1899). 
t  Zeit.  phys.  Chem.,  26,  493  (1898). 


382  THEORETICAL  CHEMISTRY 

and-sixty-thousandth  of  a  formula  weight  of  mannite  per  liter. 
Such  a  substance  is  termed  a  negative  catalyst. 

Mechanism  of  Catalysis.  As  to  the  cause  of  catalytic  action 
very  little  is  known.  In  fact  it  is  more  reasonable  to  suppose  that 
the  mechanism  of  catalysis  varies  with  the  nature  of  the  reaction 
and  the  nature  of  the  catalyst,  than  to  conceive  all  catalytic  effects 
to  be  traceable  to  a  common  origin.  One  of  the  earliest  hypotheses 
as  to  the  mechanism  of  catalysis  was  put  forward  by  Liebig.  He 
suggested  that  the  catalyst  sets  up  intramolecular  vibrations  which 
assist  chemical  reaction.  The  vibration  theory  was  gradually 
abandoned  as  its  inadequacy  came  to  be  recognized.  Of  the  many 
explanations  which  have  been  offered  to  account  for  catalytic 
acceleration,  that  involving  the  formation  of  hypothetical  inter- 
mediate compounds  with  the  catalyst  has  been  accepted  with  the 
greatest  favor.  Thus,  if  a  reaction  represented  by  the  equation 

A  +  B  =  AB, 

takes  place  very  slowly  under  ordinary  conditions,  it  is  possible 
to  accelerate  its  velocity  by  the  addition  of  an  appropriate  cat- 
alyst C.  According  to  the  theory  of  intermediate  compounds, 
the  catalyst  is  supposed  to  act  in  the  following  manner  :  — 

(1)  A  +  C  =  AC, 

(2)  AC  +  B  =  AB  +  C. 

As  will  be  seen,  the  catalyst  is  regenerated  in  the  second  stage 
of  the  reaction.  In  1806  Clement  and  Desormes  suggested  that 
the  action  of  nitric  oxide  in  promoting  the  oxidation  of  sulphur 
dioxide  in  the  manufacture  of  sulphuric  acid  was  purely  catalytic. 
As  is  well  known,  the  rate  of  the  reaction  represented  by  the 
equation 

2  SO2  +  O2  =  2  SO3, 

is  very  slow.  The  accelerating  action  of  nitric  oxide  on  the 
reaction  may  be  represented  in  the  following  manner :  — 

(1)  2NO  +  O2  =  2NO2, 
and 

(2)  SO2  +  NO2  =  SO3  +  NO. 


CHEMICAL  KINETICS  383 

This  explanation,  first  offered  by  Clement  and  Desormes,  is  still 
regarded  as  the  most  plausible  explanation  of  the  part  played  by 
the  oxides  of  nitrogen  in  the  synthesis  of  sulphuric  acid.  It  is 
apparent  that  this  so-called  explanation  is  far  from  complete.  In 
fact,  it  must  be  admitted  that  we  have  no  adequate  explanation 
of  the  phenomenon  of  catalysis.  When  we  are  able  to  answer 
the  question — "Why  does  a  chemical  reaction  take  place  ?"- 
then  we  may  be  able  to  explain  the  accelerating  and  retarding 
influences  of  certain  foreign  substances  on  the  speed  of  reactions. 
Ostwald  likens  the  action  of  a  catalyst  to  that  of  a  lubricant  on  a 
machine  —  it  helps  to  overcome  the  resistance  of  the  reaction. 
If  the  velocity  of  a  reaction  is  represented  by  an  equation  similar 
to  that  expressing  Ohm's  law,  we  have 

,  driving  force 

velocity  of  reaction  = •—       — 

resistance 

The  driving  force  is  the  same  thing  as  the  free  energy  or  chemical 
affinity  of  the  reacting  substances;  of  the  resistance  we  know 
practically  nothing.  The  velocity,  according  to  the  above  expres- 
sion, can  be  increased  in  either  of  two  ways,  viz.,  (I)  by  increas- 
ing the  driving  force,  or  (2)  by  diminishing  the  resistance.  It  is 
inconceivable  that  a  catalyst  can  exert  any  effect  upon  the  chem- 
ical affinity  of  the  reacting  substances,  so  that  we  are  forced  to 
conclude  that  its  action  must  be  confined  to  lessening  the 
resistance.* 

PROBLEMS. 

1.  When  a  solution  of  dibromsuccinic  acid  is  heated,  the  acid  decom- 
poses into  brom-maleic  acid  and  hydrobromic  acid  according  to  the 
equation 

CHBr.COOH   CH-COOH 
|  +  HBr. 

CHBr-COOH   CBr-COOH 

*  For  an  excellent  review  of  the  subject  of  catalysis  the  student  is  advised 
to  consult  "Die  Lehre  von  der  Reaktionsbeschleunigung  durch  Fremdstoffe, " 
by  W.  Herz.  Ahrens'  "Sammlung  chemischer  and  chemiseh-technischer 
Vortraege.'5 


384 


THEORETICAL  CHEMISTRY 


At  50°  the  initial  titre  of  a  definite  volume  of  the  solution  was  T0  = 
10.095  cc.  of  standard  alkali.  After  t  minutes  the  titre  of  the  same 
volume  of  solution  was  Tt  cc.  of  standard  alkali. 

t  0  214  380 

Tt  10.095  10.37  10.57 

(a)  Calculate  the  velocity-constant  of  the  reaction. 

(b)  After  what  time  is  one-third  of  the  dibromsuccinic  acid  decom- 
posed? Ans.   (a)  0.000260;  (b)  1559  minutes. 

2.  From  the  following  data  show  that  the  decomposition  of  H202  in 
aqueous  solution  is  a  unimolecular  reaction :  — 

Time  in  minutes  0  10  20 

n  22. 8          13.8  8.25cc. 

n  is  the  number  of  cubic  centimeters  of  potassium  permanganate  required 
to  decompose  a  definite  volume  of  the  hydrogen  peroxide  solution. 

3.  In  the  saponification  of  ethyl  acetate  by  sodium  hydroxide  at  10°, 
y  cc.  of  0.043  molar  hydrochloric  acid  were  required  to  neutralize  100  cc. 
of  the  reaction  mixture  t  minutes  after  the  commencement  of  the  reaction. 

t  0  4.89          10.37          28.18        infinity 

y  61.95          50.59          42.40          29.35        14.92 

Calculate  the  velocity-constant  when  the  concentrations  are  expressed 
in  mols  per  liter.  Ans.  Mean  value  of  k  =  2.38. 

4.  The  velocity-constant  of  formation  of  hydriodic  acid  from  its  ele- 
ments is  0.00023;   the  equilibrium  constant  at  the  same  temperature  is 
0.0157.     What  is  the  velocity-constant  of  the  reverse  reaction? 

Ans.  0.0146. 

5.  Determine  the  order  of  the  following  reaction :  — 

6  FeCl2  +  KC103  +  6  HC1  =  6  FeCl3  +  KC1  +  3  H20. 
When  the  initial  concentration  of  the  reacting  substances  is  0.1,  the 
changes  in  concentration  at  successive  times  are  as  follows:  — 


Time  (minutes). 

Change  in  Con- 
centration. 

5 

0.0048 

15 

0.0122 

35 

0.0238 

60 

0.0329 

110 

0.0452 

170 

0.0525 

Ans.  Third  order. 


CHAPTER  XVIII. 
ELECTRICAL  CONDUCTANCE. 

Historical  Introduction.  In  a  book  of  this  character  it  is 
impossible  to  give  anything  like  a  complete  historical  sketch  of 
electrochemistry.  Before  entering  upon  an  outline  of  this  inter- 
esting division  of  theoretical  chemistry,  however,  it  is  desirable 
to  consider  very  briefly  a  few  of  the  theories  which  have  played  a 
prominent  part  in  the  development  of  our  modern  views  concern- 
ing electrochemical  phenomena.  While  the  early  observations  of 
Beccaria  and  others  pointed  to  the  probability  of  the  existence 
of  some  relation  between  chemical  and  electrical  phenomena,  it 
was  not  until  the  beginning  of  the  nineteenth  century  that  the 
science  of  electrochemistry  had  its  birth.  The  epoch-making 
discovery  by  Volta  of  a  means  of  obtaining  electrical  energy  from 
chemical  energy,  gave  the  initial  impulse  to  all  the  brilliant  dis- 
coveries and  investigations  upon  which  the  modern  science  of 
electrochemistry  is  based.  The  apparatus  devised  by  Volta, 
known  as  the  voltaic  pile,  consisted  of  disks  of  zinc  and  silver 
placed  alternately  over  one  another,  the  silver  disk  of  one  pair 
being  separated  from  the  zinc  disk  of  the  next  by  a  piece  of 
blotting  paper  moistened  with  brine.  Such  a  pile,  if  composed 
of  a  sufficient  number  of  pairs  of  disks,  will  produce  electricity 
enough  to  give  a  shock,  if  the  top  and  bottom  disks,  or  wires 
connected  with  them,  be  touched  with  the  moist  fingers.  This 
discovery  placed  in  the  hands  of  the  investigator  a  source  of 
electricity  by  means  of  which  experiments  could  be  performed 
which  had  hitherto  been  impossible.  Shortly  after  the  discovery 
of  the  voltaic  pile,  Nicholson  and  Carlisle  *  effected  the  decom- 
position of  water,  and  Davy  f  isolated  the  alkali  metals.  As 
a  result  of  these  experiments,  Davy  was  led  to  formulate  his 

*  Nich.  Jour.,  4,  179  (1800). 

t  Ibid.,  4,  275,  326  (1800);  Gilb.  Ann.,  7,  114  (1801). 
385 


386  THEORETICAL  CHEMISTRY 

electrochemical  theory.  According  to  this  theory,  the  atoms  of 
different  substances  acquire  opposite  electrical  charges  by  con- 
tact, and  thus  mutually  attract  each  other.  If  the  differences 
between  the  charges  are  small,  the  attraction  will  be  insufficient 
to  cause  the  atoms  to  leave  their  former  positions;  if  it  is  great, 
a  rearrangement  of  the  atoms  will  occur  and  a  chemical  com- 
pound will  be  formed.  In  terms  of  this  theory,  electrolysis  con- 
sists hi  a  neutralization  of  the  charges  upon  the  atoms. 

The  theory  of  Davy  was  soon  superseded  by  that  of  Berzelius.* 
According  to  the  latter  theory,  every  atom  is  charged  with  both 
kinds  of  electricity  which  exist  upon  the  atoms  hi  a  polar  arrange- 
ment, the  electrical  behavior  of  the  atom  being  determined  by  the 
kind  of  electricity  which  is  in  excess.  Chemical  attraction  is  merely 
the  electrical  attraction  of  oppositely-charged  atoms.  Since  each 
atom  is  endowed  with  both  positive  and  negative  electrification, 
one  charge  being  in  excess,  it  follows  that  the  compound  formed 
by  the  union  of  two  or  more  atoms  will  be  positively  or  negatively 
charged  according  to  whichever  charge  remains  unneutralized  after 
the  atoms  have  combined.  Two  compounds,  the  one  charged  pos- 
itively and  the  other  negatively,  may  thus  in  turn  combine,  a 
more  complex  compound  being  formed.  Shortly  after  Berzelius 
formulated  his  theory,  it  became  the  subject  of  much  discussion 
and  was  severely  criticized.  Thus,  it  was  pointed  out  that  if 
chemical  combination  results  from  the  neutralization  of  oppo- 
sitely-charged atoms,  then  as  soon  as  the  charges  have  become 
equalized,  there  no  longer  exists  any  attractive  force  and  the  com- 
pound must  again  decompose.  This  objection  was  easily  overcome 
by  assuming  that  as  soon  as  the  union  between  the  atoms  is 
broken,  they  again  acquire  their  original  charges  and,  in  conse- 
quence, recombine.  In  other  words,  a  chemical  compound  is  to 
be  regarded  as  existing  in  a  state  of  unstable  equilibrium.  An- 
other, and  apparently  insurmountable,  objection  to  the  theory 
resulted  from  the  exceptions  presented  by  acetic  acid  and  some  of 
its  substitution  products. 

According  to  the  theory  of  Berzelius,  chemical  combination  is 
entirely  dependent  upon  the  nature  of  the  electrical  charges  resid- 
*  Gilb.  Ann.,  27,  270  (1807). 


ELECTRICAL   CONDUCTANCE*  387 

ing  on  the  atoms.  From  this  statement  it  follows  that  the  prop- 
erties of  a  chemical  compound  must  be  a  function  of  the  electrical 
charges  upon  the  atoms  of  its  constituents.  It  was  shown  that 
when  the  three  hydrogen  atoms  of  the  methyl  group  in  acetic 
acid  are  successively  replaced  by  chlorine,  the  chemical  properties 
of  the  original  substance  are  not  materially  altered.  According 
to  Berzelius,  the  three  hydrogen  atoms  are  positively  charged 
while  the  three  chlorine  atoms  are  negatively  charged.  That 
three  negative  charges  could  be  substituted  for  three  positive 
charges  in  acetic  acid  without  producing  a  more  marked  change 
in  its  properties,  could  not  be  satisfactorily  accounted  for  by  the 
theory.  This  criticism  was  for  a  long  time  considered  as  an 
insuperable  barrier  to  the  acceptance  of  the  theory.  Shortly 
before  the  close  of  the  nineteenth  century,  J.  J.  Thomson  *  showed 
that  this  objection  has  little  or  no  weight.  When  hydrogen  gas 
is  electrolyzed  in  a  vacuum-tube  and  the  spectra  at  the  two  elec- 
trodes are  compared,  Thomson  found  them  to  differ  widely. 
From  this  he  concluded  that  the  molecule  of  hydrogen  gas  is  in 
all  probability  made  up  of  positively-  and  negatively-charged  parts 
or  ions.  He  then  extended  his  experiments  to  the  vapors  of  cer- 
tain organic  compounds.  In  discussing  these  experiments  he 
says:  —  "In  many  organic  compounds,  atoms  of  an  electro- 
positive element,  hydrogen,  are  replaced  by  atoms  of  an  elec- 
tronegative element,  chlorine,  without  altering  the  type  of  the 
compound.  Thus,  for  example,  we  can  replace  the  four  hydrogen 
atoms  in  CH4  by  chlorine  atoms,  getting,  successively,  the  com- 
pounds CH3C1,  CH2C12,  CHC13,  and  CC14.  It  seemed  of  interest 
to  investigate  what  was  the  nature  of  the  charge  of  electricity  on 
the  chlorine  atoms  in  these  compounds.  The  point  is  of  some 
historical  interest,  as  the  possibility  of  substituting  an  electro- 
negative element  in  a  compound  for  an  electropositive  one  was 
one  of  the  chief  objections  against  the  electrochemical  theory  of 
Berzelius." 

"  When  the  vapor  of  chloroform  was  placed  in  the  tube,  it  was 
found  that  both  the  hydrogen  and  chlorine  lines  were  bright  on 
the  negative  side  of  the  plate,  while  they  were  absent  from  the 
*  Nature,  52,  451  (1895). 


388  '   THEORETICAL  CHEMISTRY 

positive  side,  and  that  any  increase  in  brightness  of  the  hydrogen 
lines  was  accompanied  by  an  increase  in  the  brightness  of  those 
due  to  chlorine.  The  appearance  of  the  hydrogen  and  chlorine 
spectra  on  the  same  side  of  the  plate  was  also  observed  in  methy- 
lene  chloride  and  in  ethylene  chloride.  Even  when  all  the 
hydrogen  in  methane  was  replaced  by  chlorine,  as  in  carbon  tetra- 
chloride,  the  chlorine  spectra  still  clung  to  the  negative  side  of 
the  plate.  The  same  point  was  tested  with  silicon  tetrachloride 
and  the  chlorine  spectrum  was  brightest  on  the  negative  side  of 
the  plate.  From  these  experiments  it  would  appear,  that  the 
chlorine  atoms  in  the  chlorine  derivatives  of  methane  are  charged 
with  electricity  of  the  same  sign  as  the  hydrogen  atoms  they 
displace. " 

Electrical  Units.  In  1827,  Dr.  G.  S.  Ohm  enunciated  his  well- 
known  law  of  electrical  conductance,  viz. :  —  The  strength  of  the 
electric  current  flowing  in  a  conductor  is  directly  proportional  to  the 
difference  of  potential  between  the  ends  of  the  conductor,  and  inversely 
proportional  to  its  resistance.  If  C  represents  the  strength  of  the 
current,  E  the  difference  of  potential,  and  R  the  resistance,  then 
Ohm's  law  may  be  formulated  thus :  — 

C      E 

V 

The  unit  of  resistance  is  the  ohm,  that  of  difference  of  potential 
or  electromotive  force,  the  volt,  and  that  of  current,  the  ampere. 
The  ohm  is  denned  as  the  resistance  of  a  column  of  mercury 
106.3  cm.  long  and  1  sq.  mm.  in  cross  section  at  0°  C.  The 
ampere  is  defined  as  the  current  which  will  cause  the  deposition 
of  0.001118  gram  of  silver  from  a  solution  of  silver  nitrate  in  1 
second.  The  volt  may  be  defined  as  the  electromotive  force 
necessary  to  drive  a  current  of  1  ampere  through  a  resistance  of 
1  ohm.  The  unit  of  quantity  of  electricity  is  the  coulomb,  and  is 
equivalent  to  a  current  of  1  ampere  per  second.  One  gram 
equivalent  of  any  ion  carries  96,500  coulombs,  a  quantity  of 
electricity  known  as  the  faraday  =  F.  As  has  already  been 
pointed  out,  any  form  of  energy  may  be  considered  as  the  product 
of  two  factors,  a  capacity  factor  and  an  intensity  factor. 


ELECTRICAL   CONDUCTANCE  389 

The  capacity  factor  of  electrical  energy  is  the  coulomb  while 
the  intensity  factor  is  the  volt,  i.e., 

electrical  energy  =  coulombs  X  volts. 

The  unit  of  electrical  energy,  therefore,  is  the  volt-ampere-second 
commonly  called  the  watt-second.  One  watt-second  is  the  elec- 
trical work  done  by  a  current  of  1  ampere  flowing  under  an  elec- 
tromotive force  of  1  volt  for  1  second,  and  is  equivalent  to  1  X  107 
C.G.S.  units.  The  thermal  equivalent  of  electrical  energy  may  be 
calculated  from  the  relation 

electrical  energy  in  absolute  units       ,  .    . 

.    i r-  -  =  heat  equiv.  of  elect,  energy. 

mechanical  equiv.  of  heat 


or 

1X107 


42,600  X  980.1 


=  0.2394  cal.  =  1  watt-second. 


Faraday's  Laws.  When  two  platinum  plates  or  electrodes,  one 
connected  to  the  positive  and  the  other  to  the  negative  terminal 
of  a  battery,  are  immersed  in  a  solution  of  sodium  chloride,  it 
will  be  found  that  hydrogen  is  immediately  evolved  at  the  nega- 
tive electrode  and  oxygen  at  the  positive  electrode.  If  the  salt 
solution  is  previously  colored  with  a  few  drops  of  a  solution  of 
litmus  it  will  be  observed  that  the  portion  of  the  solution  in  the 
neighborhood  of  the  positive  electrode  will  turn  red,  indicating 
the  formation  of  an  acid,  while  that  in  the  neighborhood  of  the 
negative  electrode  will  turn  blue,  showing  the  formation  of  a 
base.  The  same  changes  will  take  place  whether  the  electrodes 
are  placed  near  together  or  far  apart,  and  furthermore,  the  evolu- 
tion of  gas  and  the  change  in  color  at  the  electrodes  commences 
as  soon  as  the  circuit  is  closed.  The  study  of  these  phenomena 
led  Faraday  *  to  the  conclusion,  that  when  an  electric  current 
traverses  a  solution,  there  occurs  an  actual  transfer  of  matter, 
one  portion  travelling  with  the  current  and  the  other  portion 
moving  in  the  opposite  direction.  At  the  suggestion  of  the  philol- 
ogist Whewell,  Faraday  termed  these  carriers  of  the  current,  ions 
to  wander).  He  also  called  the  electrode  connected  to 
*  Experimental  Researches,  (1834). 


390 


THEORETICAL  CHEMISTRY 


the  positive  terminal  of  the  battery,  the  anode,  (ava.  =  up  and 
oSos  =  way),  and  the  electrode  connected  to  the  negative  terminal 
t.he  cathode,  (Kara  =  down  and  680$  =  way).  The  ions  which 
move  toward  the  anode  he  called  anions,  while  those  which  migrate 
toward  the  cathode  he  called  cations.  The  whole  process  he 
termed  electrolysis.  The  question  of  the  relationship  between  the 
amount  of  electrolysis  and  the  quantity  of  electricity  passing 
through  a  solution  was  investigated  by  Faraday.  As  a  result  of 
his  experiments  he  enunciated  the  following  laws  which  are  com- 
monly known  as  the  laws  of  Faraday :  — 

(1)  For  .the  same  electrolyte,  the  amount  of  electrolysis  is  propor- 
tional to  the  quantity  of  electricity  which  passes. 

(2)  The  amounts  of  substances  liberated  at  the  electrodes  when  the 
same  quantity  of  electricity  passes  through  solutions   of  different 
electrolytes,   are  proportional   to   their  chemical  equivalents.     The 
chemical  equivalent  of  any  ion  is  equal  to  the  atomic  weight  divided 
by  its  valence.     If  the  same  quantity  of  electricity  is  passed 
through  solutions  of  hydrochloric  acid,   silver  nitrate,   cuprous 
chloride,  cupric  chloride,  and  auric  chloride,  the  relative  amounts 
of  the  different  cations  liberated  will  be  as  follows:  — 


Electrolyte. 

Chem.  Equiv.  of 
Cation. 

HC1 

H*  =  l 

AgNO3 
Cu2Cl2 

Ag'  =  10S 
Cu'  =  63.4 

CuCl2 

Cu"  =  63.4-r2 

AuCl3 

Au-  =  197^3 

The  electrochemical  equivalent  of  an  element  or  group  of  elements 
is  the  weight  in  grams  which  is  liberated  by  the  passage  of  one 
coulomb  of  electricity.  The  electrochemical  equivalents  are, 
according  to  Faraday's  second  law,  proportional  to  the  chemical 
equivalents.  The  quantity  of  electricity  necessary  to  liberate  one 
chemical  equivalent  in  grams  is  called  a  faraday.  This  is  a  very 
important  unit  in  electrochemical  calculations.  Since  one  coulomb 
liberates  0.00001036  gram  of  hydrogen,  1  -^  0.00001036  =  96,500 


ELECTRICAL   CONDUCTANCE  391 

coulombs  of  electricity  will  be  required  to  liberate  one  gram  equiv- 
alent of  hydrogen.  The  same  quantity  of  electricity  will  liberate 
35.45  X  0.00001036  =  0.000368  gram  of  chlorine,  and  108  X 
0.00001036  =  0.001118  gram  of  silver.  Or,  in  general,  since  one 
coulomb  of  electricity  liberates  0.00001036  gram  of  hydrogen,  it 
will  cause  the  liberation  of  0.00001036  w  grams  of  any  other  ele- 
ment whose  equivalent  weight  is  w. 

The  Existence  of  Free  Ions.  When  an  electrolyte  is  de- 
composed by  the  electric  ^current,  the  products  of  decomposition 
appear  at  the  electrodes.  The  fact  that  the  liberation  of  the  prod- 
ucts of  decomposition  is  independent  of  the  distance  between 
the  electrodes  caused  considerable  difficulty  in  the  early  history 
of  electrolysis.  It  was  evident  that  the  two  products  could 
hardly  be  derived  from  the  same  molecule,  but  must  come  from 
two  different  molecules.  Several  theories  were  advanced  to 
account  for  the  experimental  results.  Thus,  in  the  electrolysis  of 
water  it  was  suggested  that  the  two  gases,  hydrogen  and  oxygen, 
were  not  derived  from  the  water  but  that  electricity  itself  pos- 
sessed an  acid  character.  Grotthuss  *  was  the  first  to  propose  a 
rational  hypothesis  as  to  the  mechanism  of  electrolysis.  He 
assumed  that  when  the  electrodes  in  an  electrolytic  cell  are  con- 
nected with  a  source  of  electricity,  the  molecules  of  the  electrolyte 
arrange  themselves  in  straight  lines  between  the  electrodes,  the 
positive  poles  being  directed  toward  the  negative  electrode  and 
the  negative  poles  toward  the  positive  electrode.  When  elec- 
trolysis begins,  the  cation  of  the  molecule  nearest  the  cathode 
is  liberated  at  the  cathode  and  the  anion  of  the  molecule  nearest 
the  anode  is  liberated  at  the  anode.  The  anion  which  is  left 
free  near  the  cathode  then  combines  with  the  cation  of  the  next 
adjoining  molecule,  the  anion  thus  left  uncombined  uniting  with 
the  cation  of  its  nearest  neighbor,  a  similar  exchange  of  partners 
continuing  throughout  the  entire  molecular  chain.  Under  the 
lirective  influence  of  the  two  electrodes,  the  newly-grouped  mole- 
cules then  rotate  so  that  the  positive  poles  all  face  the  negative 
electrode  and  the  negative  poles  all  face  the  positive  electrode. 
The  process  is  then  repeated,  another  molecule  being  electrolyzed. 
*  Ann.  de  Chim.  [1],  58,  54  (1806). 


392  THEORETICAL  CHEMISTRY 

This  theory  of  electrolysis  appears  to  have  been  accepted  by 
Faraday.  Its  inherent  defect  was  first  pointed  out  by  Grove.* 
From  his  experiments  with  the  oxy-hydrogen  cell,  which  derives 
its  energy  from  the  union  of  hydrogen  and  oxygen,  he  pointed  out 
that  a  decomposition  of  the  molecules  of  water  is  not  essential 
for  the  evolution  of  these  two  gases,  but  that  the  molecules  must 
be  already  in  a  state  of  partial  decomposition.  This  suggestion 
was  followed  up  by  Clausius.f  He  argued  that  if  an  expenditure 
of  energy  is  necessary  to  decompose  the  molecules,  electrolysis 
should  be  impossible  at  very  low  voltages.  Experiment  showed 
that  when  silver  nitrate  is  electrolyzed  between  silver  electrodes, 
decomposition  takes  place  at  voltages  which  are  much  below  the 
voltage  corresponding  to  the  energy  of  formation  of  silver  nitrate. 
In  other  words,  it  requires  very  little  energy  to  decompose  a  salt 
which  is  formed  with  the  evolution  of  a  large  amount  of  energy, 
a  result  which  is  in  contradiction  to  the  principle  of  the  conserva- 
tion of  energy.  Clausius  was  thus  forced  to  conclude  "  that  the 
supposition  that  the  constituents  of  the  molecule  of  an  electrolyte 
are  firmly  united  and  exist  in  a  fixed  and  orderly  arrangement  is 
wholly  erroneous." 

As  a  result  of  his  investigation  of  the  synthesis  of  ethyl  ether 
from  alcohol  and  sulphuric  acid,  Williamson  t  concluded  "that  in 
an  aggregate  of  the  molecules  of  every  compound,  a  constant  inter- 
change between  the  elements  contained  in  them  is  taking  place. " 
In  the  same  paper  he  writes,  "each  atom  of  hydrogen  does  not 
remain  quietly  attached  all  the  time  to  the  same  atom  of  chlorine, 
but  they  are  continually  exchanging  places  with  one  another." 
This  view  was  accepted  by  Clausius,  although  he  had  no  means  of 
determining  the  extent  to  which  the  electrolyte  was  broken  down 
or  dissociated  into  free  ions. 

In  1887,  Arrhenius  §  developed  the  views  of  Clausius  by  showing 
how  the  degree  of  dissociation  of  the  molecules  of  an  electrolyte 
can  be  deduced  from  measurements  of  the  electrical  conductance 

*  Phil.  Mag.,  27,  348  (1845). 
t  Pogg.  Ann.,  101,  338  (1857). 
t  Lieb.  Ann.,  77,  37  (1851). 
§  Zeit.  phys.  Chem.,  i,  631  (1887). 


ELECTRICAL   CONDUCTANCE 


393 


of  its  solutions,  as  well  as  from  measurements  of  osmotic  pressure 
and  freezing-point  lowering.  The  important  generalization  sum- 
marizing these  conceptions  is  known  as  the  theory  of  electrolytic 
dissociation,  to  which  reference  has  already  been  made  in  earlier 
chapters  (see  page  227). 

The  Migration  of  the  Ions.  Since  the  passage  of  a  current  of 
electricity  through  a  solution  of  an  electrolyte  causes  the  dis- 
charge of  equivalent  amounts  of  positive  and  negative  ions  at  the 
electrodes,  it  might  be  inferred  that  the  ions  all  move  with  the 
same  speed.  That  this  inference  is  incorrect,  was  first  shown  by 
Hittorf  *  as  the  result  of  his  observations  on  the  changes  in  con- 
centration of  the  solution  in  the  neighborhood  of  the  electrodes 


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0 

0 

©jo© 

Fig.  87. 

during  electrolysis.  He  showed  that  different  ions  migrate  with 
different  speeds,  and  that  the  faster  moving  ions  carry  a  greater 
proportion  of  the  current  than  the  slower  moving  ions.  The 
effect  of  unequal  ionic  velocities  on  the  concentrations  of  the 
solutions  around  the  electrodes  is  clearly  shown  by  the  ac- 
companying diagram  (Fig.  87)  due  to  Ostwald.  The  anode  and 

*  Pogg.  Ann.,  89,  177;  98,  1;  103,  1;  106,  337,  513  (1853-1859). 


394  THEORETICAL  CHEMISTRY 

cathode  in  an  electrolytic  cell  are  represented  by  the  vertical  lines 
A  and  C  respectively.  The  cell  is  divided  into  three  compart- 
ments by  means  of  porous  diaphragms,  represented  by  the  ver- 
tical dotted  lines.  The  cations  are  represented  by  dots  (")  and 
the  anions  by  dashes  (')•  Before  the  current  passes  through  the 
cell,  the  concentration  of  the  solution  is  uniform  throughout,  the 
conditions  being  represented  by  I.  Now  let  us  imagine  that  only 
the  anions  move  when  the  current  is  established.  The  conditions 
when  the  chain  of  anions  has  moved  two  steps  toward  the  anode 
are  shown  in  II.  Each  ion  which  has  been  deprived  of  a  partner 
is  supposed  to  be  discharged.  It  will  be  observed  that  although 
the  cations  have  not  migrated  toward  the  cathode,  yet  an  equal 
number  of  positive  and  negative  ions  are  discharged,  and  that 
while  the  concentration  in  the  anode  compartment  has  not  changed, 
the  concentration  in  the  cathode  compartment  has  diminished  to 
one-half  its  original  value. 

Let  us  now  suppose  that  both  anions  and  cations  move  with  the 
same  speed,  and  as  before,  let  each  chain  of  ions  move  two  steps 
toward  their  respective  electrodes,  as  indicated  in  III.  It  will  be 
seen  that  four  positive  and  four  negative  ions  have  been  dis- 
charged, and  that  the  concentration  of  the  electrolyte  in  the  anode 
and  cathode  compartments  has  diminished  to  the  same  extent. 
Finally,  let  us  assume  that  the  ratio  of  the  speeds  of  the  cations 
to  that  of  the  anions  is  as  3  :  2.  When  the  cations  have  moved 
three  steps  toward  the  cathode  and  the  anions  have  moved  two 
steps  toward  the  anode,  the  conditions  will  be  as  shown  in  IV. 
It  is  evident  that  five  positive  and  five  negative  ions  have  been 
discharged,  and  that  the  concentration  in  the  cathode  compart- 
ment has  diminished  by  two  molecules  while  the  concentration 
in  the  anode  compartment  has  diminished  by  three  molecules. 
It  will  be  observed  that  the  change  in  concentration  in  either  of 
the  electrode  compartments  is  proportional  to  the  speed  of  the 
ion  leaving  it.  Thus,  in  II,  the  concentration  in  the  cathode 
compartment  diminishes  while  that  in  the  anode  compartment 
remains  unchanged,  since  only  the  anion  moves.  In  like  manner, 
the  change  in  concentration  about  the  electrodes  in  III  corre- 
sponds with  the  fact  that  both  ions  migrate  at  the  same  rate. 


ELECTRICAL   CONDUCTANCE  395 

In  IV  the  ratio  of  the  change  in  concentration  in  the  cathode 
compartment  to  that  in  the  anode  compartment  is  as  2  :  3.  It 
will  be  apparent  from  these  examples,  that  the  relation  between 
the  speeds  of  the  ions  and  the  corresponding  changes  in  concen- 
tration at  the  electrodes  may  be  expressed  by  the  following  pro- 
portion :  — 

Change  in  concentration  at  anode    _  speed  of  cation 
Change  in  concentration  at  cathode       speed  of  anion 

If  the  relative  speed  of  the  cations  is  represented  by  u,  and  that 
of  the  anions  by  v,  then  the  total  quantity  of  electricity  trans- 
ported will  be  proportional  to  u  -f-  v:  of  this  total,  the  fractions 

carried  by  the  anion  and  cation  respectively,  will  be  n  =  — - — , 

u  +  v 

and     1  —  n  =  -     —  •     The  values  of  these  ratios,  n  and  I  —  n. 
u  +  v 

are  called  the  transport  numbers  of  the  anion  and  cation  respec- 
tively. It  is  apparent  from  the  diagram,  that  if  the  electrolysis 
is  not  carried  too  far,  the  concentration  of  the  solution  in  the  inter- 
mediate compartment  will  undergo  no  change.  In  order  to  deter- 
mine transport  numbers,  therefore,  it  is  simply  necessary  to 
remove  portions  of  the  solutions  in  the  immediate  vicinity  of  the 
two  electrodes  and  determine  the  concentration  of  the  electrolyte 
analytically.  The  success  of  the  experiment  depends  upon  keep- 
ing the  concentration  of  the  intermediate  compartment  unaltered. 
Experimental  Determination  of  Transport  Numbers.  Various 
forms  of  apparatus  have  been  constructed  for  the  determination 
of  transport  numbers,  among  which  one  of  the  most  satisfactory 
is  that  devised  by  Jones  and  Bassett,*  and  shown  in  Fig.  88.  It 
consists  of  two  vertical  tubes  of  wide  bore  connected  by  a  U-tube 
fitted  with  a  stop-cock.  Into  each  of  two  electrodes,  made  of 
a  suitable  metal,  is  riveted  a  short  piece  of  stout  platinum  wire, 
which  is  then  sealed  into  heavy-walled  glass  tubes.  The  exposed 
end  of  the  platinum  wire  on  the  under  side  of  each  electrode  is 
covered  with  a  drop  of  fusion  glass.  The  tubes  carrying  the 
electrodes  are  fitted  into  holes  bored  through  the  ground  glass 

*  Am.  Chem.  Jour.,  32,  409  (1904). 


396 


THEORETICAL  CHEMISTRY 


stoppers  which  close  the  right  and  left  arms  of  the  apparatus. 
Two  small  graduated  tubes  are  sealed  to  the  two  vertical  tubes 
just  below  the  stoppers.  These  tubes  allow  for  any  slight  dis- 
placement of  the  solution  due  to  expansion  or  the  formation  of 


Fig.  88. 

gas,  and  at  the  same  time  make  it  possible  to  level  the  apparatus 
accurately.  When  electrolysis  has  proceeded  far  enough,  the 
circuit  is  broken  and  the  stop-cock  closed,  thus  preventing  the 
mixing  of  the  solutions  in  the  anode  and  cathode  compartments. 
The  solutions  in  the  two  halves  of  the  apparatus  are  then  rinsed  out 
into  separate  beakers  and  the  concentration  of  each  is  determined 
analytically.  Knowing  the  initial  concentration  of  the  solution 
and  the  final  concentrations  at  the  two  electrodes,  together  with 
the  total  quantity  of  electricity  which  has  passed  through  the 
apparatus  during  the  experiment,  we  have  all  of  the  data  neces- 
sary for  the  calculation  of  the  transport  numbers  of  the  two  ions. 


ELECTRICAL   CONDUCTANCE  397 

The  following  example  will  serve  to  make  the  method  of  calcu- 
lation clear  :  —  In  an  experiment  to  determine  the  transport 
numbers  of  the  ions  of  silver  nitrate,  a  solution  containing  0.00739 
gram  of  that  salt  per  gram  of  water  was  prepared.  The  solution 
was  introduced  into  the  migration  apparatus  and,  after  inserting 
silver  electrodes,  a  small  current  was  passed  through  the  appa- 
ratus for  two  hours.  A  silver  coulometer  was  included  in  the  cir- 
cuit, and  0.0780  gram  of  silver  was  deposited  by  the  current. 

This  mass  of  silver  is  equivalent  to  0.000723  gram-equivalent. 
After  the  circuit  was  broken,  the  anode  solution  was  rinsed  out  and 
its  concentration  determined  analytically.  It  was  found  to  con- 
tain 0.2361  gram  of  silver  nitrate  to  23.14  grams  of  water.  This 
amount  of  solution  contained  originally  23.14  X  0.00739  = 
0.1710  gram  of  silver  nitrate.  Thus,  the  amount  of  silver  nitrate 
in  the  anode  compartment  had  increased  by  0.2361  —  0.1710  = 
0.0651  gram  of  silver  nitrate,  or  0.000383  gram-equivalent  of 
silver.  Obviously  the  increase  in  the  concentration  of  the  nitrate 
ion  must  have  been  the  same.  The  amount  of  silver  dissolved 
from  the  anode  must  have  been  equal  to  that  deposited  in  the 
coulometer,  or  since  0.000723  gram-equivalent  of  silver  was 
deposited  and  the  actual  increase  found  was  0.000383  gram- 
equivalent,  the  difference,  0.000723  -  0.000383  =  0.000340  gram- 
equivalent,  is  the  amount  of  silver  which  migrated  away  from  the 
anode.  At  the  same  time  0.000383  gram-equivalent  of  nitrate 
ions  migrated  into  the  anode  compartment.  The  ratio  of  the 
speed  of  migration  of  the  silver  ions  to  that  of  the  nitrate  ions  is 
as  0.000340  :  0.000383.  Since  0.000723  gram-equivalent  of  silver 
ions  measures  the  total  quantity  of  electricity  transported,  the 
transport  numbers  of  the  two  ions  will  be  as  follows  :  — 


Transport  number  of  Ag  =  n  =  =  0.470, 


Transport  number  of  NO3'  =  1  -  n  =  ^       f  =  0.530. 

0.000  1  Z*j 

These  numbers  can  be  checked  by  a  similar  calculation  based  on 
the  change  in  concentration  in  the  cathode  compartment. 


398 


THEORETICAL  CHEMISTRY 


The  following  table  gives  the  transport  numbers  of  the  an  ions 
of  various  electrolytes  at  different  dilutions,  V  being  the  number 
of  liters  of  solution  containing  one  gram-equivalent  of  solute. 
The  transport  numbers  of  the  corresponding  cations  can  be  found 
by  subtracting  the  transport  numbers  of  the  anions  from  unity. 

TRANSPORT  NUMBERS  OF  ANIONS. 


7  = 

100 

50 

20 

10 

5 

2 

i 

0.5 

KC1        } 
KBr 
KI 
NH4C1    J 

NaCl 

0.506 

0.507 

0.507 
0  614 

0.508 
0  617 

0.509 
0  620 

0.513 
0  626 

0.514 
0  637 

0.515 

KNO3 

0  497 

0  496 

0  492 

0  487 

0  479 

AgNO3 

0.528 

6.528 

0  528 

0  528 

0  527 

0.519 

0  501 

0  476 

KOH 

0.735 

0.736 

HC1 

0.172 

0.172 

0.172 

0.173 

0.176 

\  BaCl2    . 

0  640 

0  657 

1  K2CO3  

0.435 

0  434 

0  413 

|CuSO4  

0.620 

0.626 

0.632 

6.643 

0.668 

0.696 

0  720 

£H2i3O4  

0.182 

0.174 

It  is  apparent  from  the  table  that  the  transport  numbers  are 
not  entirely  independent  of  the  concentration.  They  also  vary 
slightly  with  the  temperature  and  approach  the  limiting  value, 
0.5,  at  high  temperatures. 

Specific,  Molar  and  Equivalent  Conductance.  As  is  well 
known,  the  resistance  of  a  metallic  conductor  is  directly  propor- 
tional to  its  length  and  inversely  proportional  to  its  area  of  cross- 
section.  Similarly,  the  resistance  of  an  electrolyte  is  proportional 
to  the  length  and  inversely  proportional  to  the  cross-section  of 
the  column  of  solution  between  the  two  electrodes.  The  specific 
resistance  of  an  electrolyte  may  be  defined  as  the  resistance  in 
ohms  of  a  column  of  solution  one  centimeter  long  and  one  square 
centimeter  in  cross-section.  Specific  conductance  is  the  reciprocal 
of  specific  resistance.  Since  the  conductance  of  a  solution  is 
almost  wholly  dependent  upon  the  amount  of  solute  present,  it 
is  more  convenient  to  express  conductance  in  terms  of  the  molar 
or  equivalent  concentration.  The  molar  conductance  p,  is  the 


ELECTRICAL   CONDUCTANCE 


399 


conductance  in  reciprocal  ohms,  of  a  solution  containing  one  mol 
of  solute  when  placed  between  electrodes  which  are  exactly  one 
centimeter  apart.  The  equivalent  conductance  A  is  the  conduc- 
tance in  reciprocal  ohms  of  a  solution  containing  one  gram- 
equivalent  of  solute  when  placed  between  electrodes  which  are 
one  centimeter  apart.  If  K  denotes  the  specific  conductance  of  a 
solution  and  Vmj  the  volume  in  cubic  centimeters  which  contains 
one  mol  of  solute,  then 


and  in  like  manner 


A   = 


where  Ve  is  the  volume  of  solution  in  cubic  centimeters  which 
contains  one  gram-equivalent  of  solute.  The  following  table 
gives  the  specific  and  molar  conductance  of  solutions  of  sodium 
chloride  at  18°  C.:  — 


Concentration. 

Dilution. 

Sp.  Cond. 

Molar  Cond. 

1 

1,000 

0.0744 

74.4 

0.1 

10,000 

0.00925 

92.5 

0.01 

100,000 

0.001028 

102.8 

0.001 

1,000,000 

0.0001078 

107.8 

0.0001 

10,000,000 

0.00001097 

109.7 

It  will  be  observed  that  the  molar  conductance  increases  with  the 
dilution  up  to  a  certain  point  beyond  which  it  remains  nearly 
constant.  That  the  molar  conductance  should  change  but  little 
will  become  apparent  from  the  following  considerations:  - 
Imagine  a  rectangular  cell  of  indefinite  height  and  having  a  cross- 
sectional  area  of  one  square  centimeter,  and  further  assume  that 
two  opposite  walls  can  function  as  electrodes.  Let  1000  cc.  of  a 
solution  containing  one  mol  of  solute  be  introduced  into  the  cell, 
and  let  its  conductance  be  determined.  Now  let  the  solution  be 
diluted  to  2000  cc.  and  the  conductance  of  the  diluted  solution  be 
measured.  While  the  specific  conductance  of  the  diluted  solution 
is  reduced  to  one-half  of  its  original  value,  yet  since  the  electrode 


400 


THEORETICAL  CHEMISTRY 


surface  in  contact  with  the  solution  is  doubled,  owing  to  the  fact 
that  the  solution  stands  at  twice  the  original  height  in  the 
cell,  the  total  conductance  due  to  one  mol  of  solute  remains  un- 
changed. This,  of  course,  is  only  the  case  with  completely  ionized 
solutes. 

Determination  of  Electrical  Conductance.  The  determination 
of  the  electrical  conductance  of  a  solution  resolves  itself  into 
the  determination  of  its  resistance  by  a  simple  modification  of 
the  familiar  Wheatstone-bridge  method.  The  arrangement  of  the 
apparatus  for  this  method  devised  by  Kohlrausch  *  is  represented 
diagrammatically  in  Fig.  89,  where  ab  is  the  bridge  wire,  B  is  a 


Fig.  89. 

resistance  box,  and  C  is  a  cell  containing  the  solution  whose 
resistance  is  to  be  measured.  The  points  d  and  c  are  connected 
to  a  small  induction  coil  /  which  gives  an  alternating  current. 
This  is  necessary  in  order  to  prevent  polarization  which  would 
occur  if  a  direct  current  were  used.  The  use  of  the  alternating 
current  necessitates  the  substitution  of  a  telephone,  T,  for  the 
galvanometer  usually  employed  in  measuring  resistance.  The 
positions  of  the  induction  coil,  and  telephone  are  sometimes  inter- 
changed, but  the  arrangement  shown  in  the  diagram  is  to  be  pre- 
ferred, since  it  insures  a  high  electromotive  force  where  the  sliding 
*  Wied.  Ann.,  6,  145  (1879);  u,  653  (1880);  26,  161  (1885). 


ELECTRICAL   CONDUCTANCE 


401 


contact  c  touches  the  wire,  this  being  the  most  uncertain  connec- 
tion in  the  entire  arrangement.  A  small  accumulator  A,  serves 
to  operate  the  induction  coil.  In  making  a  measurement,  the 
coil  is  connected  with  the  accumulator  and  the  vibrator  adjusted 
so  that  a  high  mosquito-like  tone  is  emitted;  then  the  sliding 
contact  c  is  moved  along  the  wire  ab  until  the  sound  in  the  tele- 
phone reaches  a  minimum,  the  position  of  the  point  of  contact 
with  the  "bridge-wire  being  read  on  the  millimeter  scale  placed 
below.  According  to  the  principle  of  the  Wheatstone  bridge,  it 
follows  that 

C  =  bc 

B      ac 


Since  the  resistance  B  and  the  lengths  be  and  ac  are  known,  the 
resistance  C  can  be  calculated.  Various  types  of  conductance 
cells  are  in  use,  depending  upon  whether 
the  solution  has  a  high  or  a  low  resistance. 
The  form  shown  in  Fig.  90  is  widely  used. 
The  two  electrodes  are  made  of  platinum 
foil,  connection  with  the  mercury  in  the  two 
glass  tubes  tt  being  established  by  means  of 
two  pieces  of  stout  platinum  wire  sealed 
through  the  ends  of  these  tubes.  The  tubes  tt 
are  fastened  into  a  tight-fitting  vulcanite 
cover  so  that  the  electrodes  may  be  re- 
moved, rinsed  and  dried  without  altering 
their  relative  positions.  Before  the  cell  is 
used,  the  electrodes  must  be  coated  electro- 
lytically  with  platinum  black.  It  is  not 
necessary  to  know  the  area  of  the  elec- 
trodes or  the  distance  between  them,  since 
it  is  possible  to  determine  a  factor,  termed 
the  resistance  capacity,  by  means  of  which 
the  results  obtained  with  the  cell  can  be 
transformed  into  reciprocal  ohms.  To  this 


Kit:.  HO. 


end   the   specific  conductances  of   a  number  of  standard  solu- 
tions have  been  carefully  determined  by  Kohlrauscb;   thus,  for 


402  THEORETICAL  CHEMISTRY 

a  0.02  molar  solution  of  potassium  chloride  he  found  the  following 
values  :  — 

Ki8°  =  0.002397    and     K25°  =  0.002768, 
or 

119.85    and    A25°  =  138.4. 


Let  the  resistance  of  the  cell  when  filled  with  0.02  molar  potassium 
chloride  be  C,  then  according  to  the  principle  of  the  Wheatstone 
bridge  we  have 

r      K  bc 

L    =  £>  •  —  j 

ac 
or  denoting  the  conductance  of  the  solution  by  L,  we  obtain 

T       1  -     ac 
~C~~B^bc 

Since  the  specific  conductance  K  must  be  proportional  to  the 
observed  conductance,  we  have 


where  K  is  the  resistance  capacity  of  the  cell.     If  the  measure- 
ment is  made  at  18°  C.,  then  we  have 

0.002397  B-bc 


K 


ac 


Having  determined  the  resistance  capacity  of  the  cell  we  may 
then  proceed  to  determine  the  conductance  of  any  solution.  For 
example,  suppose  that  when  the  resistance  in  the  box  is  Bf,  the 
point  of  balance  on  the  bridge-wire  is  at  c'}  then  the  specific  con- 
ductance of  the  solution  will  be 

K>  =  K^L. 

B'bc'      , 

If  K'  is  multiplied  by  the  volume  of  the  solution,  we  obtain  the 
equivalent  conductance,  or 

A  =  KfV. 

Relative  Conductances  of  Different  Substances.  The  study 
of  the  electrical  conductance  of  various  solutes  in  aqueous  solu- 
tion, reveals  the  fact  that  electrolytes  differ  greatly  in  their  con- 
ducting power.  They  may  be  roughly  divided  into  two  classes:  - 


ELECTRICAL   CONDUCTANCE 


403 


those  with  high  conducting  power,  such  as  strong  acids,  strong 
bases,  and  salts;  and  those  with  low  conducting  power,  such  as 
ammonia  and  most  of  the  organic  acids  and  bases.  Further- 
more, the  equivalent  or  molar  conductance  increases  with  the  dilu- 
tion until  a  dilution  of  about  10,000  liters  is  reached,  beyond 
which  it  remains  constant.  The  following  table  gives  the  equiv- 
alent conductances  of  three  typical  electrolytes,  V  representing 
the  volume  of  the  solution  in  liters,  and  A  the  equivalent  con- 
ductance :  — 

HYDROCHLORIC    ACID. 


V 

A  (18°). 

0.333 

201.0 

1.0 

278.0 

10.0 

324.4 

100.0 

341.6 

1000.0 

345.5 

SODIUM  HYDROXIDE. 


V 

A  (18°). 

0.333 

100.7 

1.0 

149.0 

10.0 

170.0 

100.0 

187.0 

500.0 

186.0 

POTASSIUM  CHLORIDE. 


V 

A  (18°). 

0.333 

82.7 

1.0 

91.9 

10.0 

104.7 

100.0 

114.7 

1,000.0 

119.3 

10,000.0 

120.9 

404 


THEORETICAL  CHEMISTRY 


The  curves  shown  in  Fig.  91  are  plotted  from  the  data  of  the 
foregoing  table,  and  bring  out  very  clearly  the  differences  in  con- 
ducting power  possessed  by  the  three  electrolytes. 

In  general  the  conductance  of  pure  liquids  is  small.  Thus,  the 
specific  conductance  of  pure  water  at  18°  is  approximately  1  X  10~6 


Dilution,  V 
Fig.  91. 

reciprocal  ohms  and,  as  Walden  *  has  shown,  the  specific  conduct- 
ance of  a  number  of  other  solvents  is  of  the  same  order  as  that 
for  water.  Mixtures  of  two  liquids,  each  of  which  is  practically 
non-conducting,  may  have  a  conductance  differing  but  little  from 
that  of  the  two  components;  or  the  mixture  may  have  a  very 
high  conductance.  For  example,  the  conductance  of  a  mixture 

*  Zeit.  phys.  Chem.,  46,  103  (1903). 


ELECTRICAL   CONDUCTANCE  405 

of  water  and  ethyl  alcohol  is  of  the  same  order  of  magnitude  as 
that  of  the  two  components,  while  on  the  other  hand,  a  mixture 
of  water  and  sulphuric  acid,  each  of  which  in  the  pure  state  is 
practically  a  non-conductor,  has  great  conducting  power.  The 
variation  of  the  specific  conductance  of  mixtures  of  water  and 
sulphuric  acid  is  represented  in  Fig.  92,  the  concentrations  of  sul- 
phuric acid  being  plotted  on  the  axis  of  abscissae  and  the  specific 
conductances  on  the  axis  of  ordinates.  It  appears  that  as  the 


30         40          60         60          70         80          90        100        110 
P&r  Cent  Sulphuric  Add 

Fig.  92. 

concentration  of  the  sulphuric  acid  increases,  the  specific  conduct- 
ance of  the  mixture  increases  until  30  per  cent  of  acid  is  present, 
beyond  which  point  it  gradually  diminishes.  When  pure  sul- 
phuric acid  is  present  the  value  of  the  specific  conductance  is 
practically  zero.  On  dissolving  sulphur  trioxide  in  the  pure  acid, 
the  specific  conductance  increases  slightly  to  a  maximum  and  then 
falls  rapidly  to  zero.  There  is  a  minimum  in  the  curve  corre- 
sponding to  about  85  per  cent  of  acid,  a  concentration  which 


406 


THEORETICAL  CHEMISTRY 


corresponds  almost  exactly  with  the  hydrate  H2SO4.H2O.  Why 
some  liquid  mixtures  should  have  marked  conducting  power  and 
others  hardly  any,  it  is  difficult  to  explain.  Many  fused  salts, 
such  as  silver  nitrate  and  lithium  chloride,  are  excellent  conductors 
and  are  thus  exceptions  to  the  general  rule,  that  pure  substances 
belonging  to  the  second  class  of  conductors  possess  little  conduct- 
ing power. 

The  Law  of  Kohlrausch.  The  electrical  conductance  of  solu- 
tions was  systematically  investigated  by  Kohlrausch  who  showed 
that  the  limiting  value  of  the  equivalent  conductance,  which  may 
be  represented  by  A«,  is  different  for  different  electrolytes  and 
may  be  considered  as  the  sum  of  two  independent  factors,  one  of 
which  refers  to  the  cation  and  the  other  to  the  anion.  This  experi- 
mental result  is  commonly  known  as  the  law  of  Kohlrausch. 

The  limiting  value  of  the  equivalent  conductance  is  reached 
when  the  molecules  are  completely .  broken  down  into  ions,  and 
under  these  conditions  the  whole  of  the  electrolyte  participates 
in  conducting  the  current.  The  accompanying  table,  giving  the 
equivalent  conductances  at  infinite  dilution  of  several  binary 
electrolytes,  illustrates  the  truth  of  the  law  of  Kohlrausch. 


EQUIVALENT  CONDUCTANCES  AT  INFINITE  DILUTION. 


K 

Na 

Li 

NH4 

H 

Ag 

Cl 

123 

103 

95 

122 

353 

NO3 

118 

98 

350 

109 

OH 

228 

201 

C1O3    

115 

103 

C2H302  

94 

73 

83 

The  differences  between  two  corresponding  sets  of  numbers  in 
the  same  vertical  column,  and  of  any  two  corresponding  sets  of 
numbers  in  the  same  horizontal  row  will  be  found  to  be  nearly 
equal.  This  could  only  occur  when  the  limiting  conductance  is 
the  sum  of  two  entirely  independent  quantities.  Each  ion 
invariably  carries  the  same  charge  of  electricity  and  moves  with 


ELECTRICAL   CONDUCTANCE  407 

its  own  velocity  quite  independent  of  the  nature  of  its  compan- 
ion ion.     Therefore,  at  infinite  dilution,  we  have 


in  which  lc  and  la  are  the  equivalent  conductances  of  the  ions  of 
the  electrolyte  at  infinite  dilution.     From  this  it  follows  that 


and 


»-^k— t 


i^^kl^ij 


or 

lc  =  nAoo, 
and 

la  =  (1  —  n)A<». 

Thus,  the  equivalent  conductance  of  silver  nitrate  at  infinite  dilu- 
tion at  18°  is  115.5,  while  n  =  0.518  and  1  -  n  =  0.482;  there- 
fore 

lc  =  0.518  X  115.5  =  60.8, 
and 

lc  =  0.482  XI  15.5  =  55.7; 

or  one  gram-equivalent  of  silver  ions  possesses  a  conductance  of 
60.8  when  placed  between  electrodes  one  centimeter  apart  and 
large  enough  to  contain  between  them  the  entire  volume  of  solu- 
tion in  which  the  Ag*  ions  exist;  and  one  gram-equivalent  of 
NO3'  ions  under  the  same  conditions  have  a  conductance  equal 
to  55.7. 

The  values  of  the  ionic  conductances  at  infinite  dilution  remain 
constant  in  all  solutions  in  the  same  solvent  at  the  same  temper- 
ature, so  that  it  is  possible  to  calculate  the  equivalent  conductance 
for  any  substance  at  infinite  dilution. 

In  the  subjoined  table  are  given  the  ionic  conductances  of 
various  ions  at  18°  and  infinite  dilution,  together  with  their  temper- 
ature coefficients. 


408  THEORETICAL  CHEMISTRY 

IONIC  CONDUCTANCES  AT  INFINITE  DILUTION. 


Ion. 

ie 

Temp.  Coeff  . 

Li'.  . 

33.44 

0.0265 

Na"  

43.55 

0.0244 

KV. 

64.67 

0.0217 

Rb'.. 

67.6 

0.0214 

CsV. 

68.2 

0.0212 

NH4'.. 

64.4 

0.0222 

IT  

66.0 

0.0215 

A.g' 

54  02 

0  0229 

F' 

46  64 

0.0238 

or.. 

65  44 

0.0216 

Br'. 

67  63- 

0.0215 

['.. 

66.40 

0.0213 

3CN'.. 

56.63 

0.0211 

D1O3'.. 

55.03 

0.0215 

tO3'.. 

33.87 

0.0234 

NO3' 

61  78 

0  0205 

H' 

318  0 

3H' 

174  0 

j  Zn".  .  .  . 

45  6 

0  0251 

&  Me". 

46  0 

0  0256 

[Ba".. 

56  3 

0  0238 

fc  Pb"  .  . 

61.5 

0  0243 

tSO4".. 

68.7 

0.0227 

fcCO,"  

70.0 

0.0270 

In  the  case  of  weak  electrolytes  the  value  of  A»  cannot  be 
determined  directly  from  conductance  measurements,  since  before 
the  limiting  value  is  reached,  the  solution  has  become  so  dilute  as 
to  render  accurate  measurements  of  the  specific  conductance 
impossible.  The  law  of  Kohlrausch  enables  us  to  get  around 
this  difficulty.  Thus,  the  value  of  A«>  for  acetic  acid  must  be 
equal  to  the  sum  of  the  conductances  of  the  H*  and  CH3COO' 
ions.  The  conductance  of  the  H*  ion  at  18°  is,  according  to 
the  preceding  table,  318.  The  value  of  the  conductance  of  the 
CH3COO'  ion  can  be  determined  from  the  conductance  of  sodium 
acetate  at  infinite  dilution,  A*  for  this  salt  being  78.1  at  18°. 
Since  the  ionic  conductance  of  the  Na*  ion  is  43.55  at  18°,  it 
follows  that  the  conductance  of  the  CH3COO'  ion  must  be  78.1  — 
43.55  =  34.55.  Therefore,  for  acetic  acid  we  have 

A*.  =  lc  +  la  =  318  +  34.55  =  352.55  at  18°. 


ELECTRICAL   CONDUCTANCE 


409 


Bredig  *  has  shown  that  the  ionic  conductance  of  elementary 
ions  is  a  periodic  function  of  the  atomic  weight.     When  the  ionic 


80-- 


60- • 


.2 


Ba 


Cd 


80  ,120 

Atomic  Wj3igh.t> 

TFig.93. 


200 


conductances  are  plotted  as  ordinates  against  the  atomic  weights 
as  abscissae,  the  curve  shown  in  Fig.  93  is  obtained.  A  glance  at 
the  curve  shows  the  periodic  nature  of  the  relation. 

Absolute  Velocity  of  the  Ions.  Thus  far  we  have  considered 
only  the  relative  velocities  of  the  ions  and  their  conductances; 
we  now  proceed  to  the  consideration  of  their  absolute  velocities 
in  centimeters  per  second. 

Let  a  current  of  electricity  pass  through  a  centimeter  cube  of  a 
solution  of  a  binary  electrolyte.  If  the  solution  contains  m  mols 
of  solute  per  liter,  then  ra/1000  will  be  the  number  of  mols  in  the 
centimeter  cube.  The  charge  on  either  the  cation  or  the  anion 


s 


,  where  F  =  96,540  coulombs.    If  C  represents  the  total 


current,  we  have 


m 


1000 


(Ic  +  U 


Zeit.  phys.  Chem.,  13,  242  (1894). 


410  THEORETICAL  CHEMISTRY 

since  the  current  is  the  charge  which  passes  through  one  face  of 
the  cube  in  one  second.  In  a  centimeter  cube,  the  current  is 
equal  to  the  product  of  the  specific  conductance  and  the  difference 
of  potential  E,  the  latter  being  numerically  equal  to  the  potential 
gradient,  the  distance  between  the  electrodes  being  one  centi- 
meter. Hence,  we  have 

1000  KE  =  Fm  (lc  +  la). 

HE  is  expressed  in  volts  and  K  in  reciprocal  ohms,  lc  and  la  will  be 
expressed  in  centimeters  per  second,  for  on  passing  to  absolute 
electromagnetic  units,  we  have 

1000  (K  X  10~9)  (E  X  108) 

(F  X  10-1)  "».»  +  «, 

or 


where  lc  and  la  are  the  ionic  velocities  for  unit  potential  gradient  — 
1  volt  per  centimeter. 
From  this  it  follows  that 


The  equivalent  conductance  of  a  0.0001  molar  solution  of  potas- 
sium chloride  at  18°  is  128.9;   the  total  velocity  of  the  two  ions 

is  then, 

IOQ  q 

=  0.001345  cm.  per  sec. 


This  total  velocity  is  made  up  of  the  two  individual  ionic  velocities. 
The  transport  numbers  of  the  two  ions,  K"  and  Cl',  are  respectively 
0.493  and  0.507.  Hence  the  absolute  velocities  of  the  ions,  ex- 
pressed in  centimeters  per  second,  in  a  0.0001  molar  solution  of 
potassium  chloride  at  18°  are  as  follows  :  — 

u  =  0.001345  X  0.493  =  0.00066  cm.  per  sec., 
and 

v  =  0.001345  X  0.507  =  0.00068  cm.  per  seCc 


ELECTRICAL   CONDUCTANCE 


411 


The  absolute  velocities  of  some  of  the  more  common  ions  at  18° 
are  given  in  the  following  table :  — 

ABSOLUTE   IONIC  VELOCITIES. 


Ion. 

Velocity. 

Ion. 

Velocity. 

K* 

cm.  per  sec. 
0  00066 

H* 

cm.  per  sec. 

0  00320 

NH4' 

0  00066 

Cl' 

0  00069 

Na"   ...    . 

0  00045 

NO3' 

0  00064 

Li*  

0  00036 

C1O3' 

0  00057 

Ag*  

0  00057 

OH' 

0  00181 

Cr2O7"  

0.000473 

Cu" 

0  00031 

The  velocities  of  certain  ions  have  been  determined  directly. 
Thus,  the  velocity  of  the  hydrogen  ion  was. measured  by  Lodge  * 
in  the  following  manner:  —  The  tube  B,  Fig.  94,  40  cm.  long  and 
8  cm.  in  diameter,  was  graduated  and  bent  at  right  angles  at  the 


Fig.  94. 

ends.  This  was  filled  with  an  aqueous  solution  of  sodium  chloride 
in  gelatine,  colored  red  by  the  addition  of  an  alkaline  solution  of 
phenolphthalein.  When  the  contents  of  the  tube  had  gelatinized, 
the  latter  was  placed  horizontally,  connecting  two  beakers  filled 
with  dilute  sulphuric  acid  as  shown  in  the  diagram.  A  current 
of  electricity  was  passed  from  one  electrode  A  to  the  other  elec- 
trode C. 

The  hydrogen  ions  from  the  anode  vessel  were  thus  carried  along 
the  tube,  and  discharged  the  red  color  of  the  phenolphthalein  as 
they  migrated  toward  the  cathode.     In  this  manner  the  velocity 
*  Brit.  Assoc.  Report,  p.  393  (1886). 


412  THEORETICAL  CHEMISTRY 

of  the  hydrogen  could  be  observed  under  a  known  potential  gra- 
dient. The  observed  and  calculated  values  agree  excellently.  It 
was  shown  that  the  velocity  of  the  hydrogen  ions  suffered  almost 
no  retardation  from  the  high  viscosity  of  the  gelatine  solution. 
Whetham,*  in  his  experiments  on  ionic  velocity,  employed  two 
solutions  one  of  which  possessed  a  colored  ion,  the  progress  of  the 
latter  being  observed  and  its  velocity  determined  under  unit 
potential  gradient.  For  example,  consider  the  boundary  line 
between  two  equally  dense  solutions  of  the  electrolytes  AC  and 
BC,  C  being  a  colorless  and  A  a  colored  ion.  When  a  current 
passes  through  the  boundary  between  the  two  electrolytes,  the 
anion  C  will  migrate  toward  the  positive  electrode  while  the  two 
cations,  A  and  B,  will  migrate  toward  the  negative  electrode 
and  the  color  boundary  will  move  with  the  current,  its  speed  being 
equal  to  that  of  the  colored  ion  A .  In  this  way  Whetham  measured 
the  absolute  velocities  of  the  ions,  Cu",  Cr2O7",  and  Cl'.  Ionic 
velocities  have  also  been  determined  by  Steele  f  who  observed 
the  change  in  the  index  of  refraction  of  the  solution  as  the  ions 
migrated.  The  accompanying  table  gives  a  comparison  of  the 
calculated  and  observed  velocities  of  some  of  the  ions. 


Ion. 

Velocity  (obs.). 

Velocity  (calc.). 

H*  . 

cm.  per  sec. 
0.0026 

cm.  per  sec. 
0  0032 

Cu"  

0.00029 

0.00031 

cr  

0.00058 

0  .  00069 

Cr2O7" 

0  00047 

0  000473 

Conductance  and  lonization.  We  have  already  seen  that 
solutions  of  strong  acids,  strong  bases  and  salts  exert  abnormally- 
great  osmotic  pressures.  According  to  the  molecular  theory,  this 
abnormal  osmotic  activity  has  been  ascribed  to  the  presence  in 
the  solutions  of  a  greater  number  of  dissolved  particles  than  would 
be  anticipated  from  the  simple  molecular  formulas  of  the  solutes. 
The  ratio  of  the  observed  to  the  theoretical  osmotic  pressure  was 
represented,  according  to  van't  Hoff,  by  the  factor  "i." 

*  Phil.  Trans.  A.,  184,  337  (1893);   196,  507  (1895). 
t  Phil.  Trans.  A.,  198,  105  (1902). 


ELECTRICAL   CONDUCTANCE  413 

In  1887,  Arrhenius  showed  that  there  is  an  intimate  connection 
between  electrical  conductance  and  abnormal  osmotic  activity, 
only  those  solutions  conducting  the  electric  current  which  exert 
abnormally-high  osmotic  pressures.  It  had  already  been  pointed 
out  by  Kohlrausch,  that  the  equivalent  conductance  of  a  solution 
increases  at  first  with  the  dilution  and  then  ultimately  becomes 
constant.  Arrhenius  explained  this  behavior  by  assuming  that 
the  molecules  of  the  solute  are  dissociated  into  ions,  the  con- 
ductance of  the  solution  being  solely  dependent  upon  the  number 
of  ions  present.  The  dissociation  increases  with  the  dilution  until 
finally,  when  the  equivalent  conductance  has  reached  its  maximum 
value,  it  is  complete,  the  molecules  of  solute  being  entirely  broken 
down  into  ions.  This  theory  of  Arrhenius,  known  as  the  theory 
of  electrolytic  dissociation,  is  based,  as  has  been  pointed  out, 
upon  the  views  advanced  by  Clausius.  Arrhenius  showed  how 
the  degree  of  dissociation  of  an  electrolyte  can  be  calculated 
from  the  electrical  conductance  of  its  solutions.  According  to  the 
theory  of  electrolytic  dissociation,  the  conductance  of  a  solution 
is  dependent  upon  the  number  of  ions  present  in  the  solution, 
upon  their  charges,  and  upon  their  velocities.  Since  the  electric 
charges  carried  by  equivalent  amounts  of  the  ions  of  different 
electrolytes  are  equal,  and  since  the  velocities  of  the  ions  for  the 
same  electrolyte  are  practically  independent  of  the  dilution  of  the 
solution,  it  follows  that  the  increase  in  equivalent  conductance 
with  dilution  must  depend  almost  wholly  upon  the  increase  in  the 
number  of  ions  present. 

The  equivalent  conductance  at  infinite  dilution  has  been  shown 
by  the  law  of  Kohlrausch  to  be 

Aoo    =lc  +  la, 

and,  therefore,  the  equivalent  conductance  at  any  dilution  v,  must 

be 

Av  =  a  (lc  +  la), 

where  a  is  the  degree  of  dissociation  of  the  electrolyte.     Dividing 
the  second  equation  by  the  first,  we  obtain 


414 


THEORETICAL  CHEMISTRY 


This  equation  enables  us  to  calculate  the  degree  of  ionization  of 
an  electrolyte  at  any  dilution,  provided  the  conductance  of  the 
solution  at  the  particular  dilution  is  known,  together  with  its 
conductance,  at  infinite  dilution.  For  example,  Av  at  18°  for  a 
molar  solution  of  sodium  chloride  is  74.3,  and  Aoo  is  110.3;  there- 
fore, a.  =  74.3  -r-  110.3  =  0.673,  or  in  a  molar  solution,  the  mole- 
cules of  sodium  chloride  are  dissociated  to  the  extent  of  67.3  per 
cent.  A  comparison  of  the  values  of  i  based  upon  conductance 
and  osmotic  data  has  already  been  given  in  the  table  on  page  230. 
Since  Av  =  a.  (lc  +  la),  we  may  also  write 


The  Dissociation  of  Water.     Water  behaves  as  a  very  weak 
binary  electrolyte,  dissociating  according  to  the  equation, 


The  specific  conductance  of  water,  purified  with  the  utmost  care, 
has  been  determined  by  Kohlrausch  and  Heydweiller.*  Their 
results  are  given  in  the  following  table  :  — 


Temperature, 
degrees. 

Specific  Conduct- 
anceX10-«. 

0 

0.014 

18 

0.040 

25 

0.055 

34 

0.084 

50 

0.170 

The  conductance  of  pure  water  at  0°  is  so  small  that  one  milli- 
meter of  it  has  a  resistance  equal  to  that  of  a  copper  wire  of  the 
same  cross-section  and  40,000,000  kilometers  in  length,  or  in 
other  words,  long  enough  to  encircle  the  earth  one  thousand  times. 
Knowing  the  specific  conductance  of  water,  its  degree  of  dissoci- 
ation can  be  easily  calculated.  The  ionic  conductances  of  the 
two  ions  of  water  at  18°  are  as  follows:  —  H"  =  318,  and  OH'  = 
*  Zeit.  phys.  Chem.,  14,  317  (1894). 


ELECTRICAL   CONDUCTANCE  415 

174.     Therefore,  the  maximum  equivalent  conductance  of  water 
should  be 

Aoo  =  318  +  174  =  492. 

The  equivalent  conductance  at  18°,  of  a  liter  of  water  between 
electrodes  1  cm.  apart  is,  according  to  the  data  of  Kohlrausch, 

0.04  X  10~6  X  103  =  0.04  X  10~3; 
therefore 

0.04  X  10~3 


492 


=  0.8  X  10~7  =  c,  the  concentration  of  the  ions, 


H*  and  OH',  in  mols  per  liter  at  18°. 

Conductance  of  Difficultly-Soluble  Salts.  In  a  saturated 
solution  of  a  difficultly-soluble  salt,  the  solution  is  so  dilute  that  in 
general  we  may  assume  complete  ionization,  or  Av  =  Aoo . 

When  this  is  the  case,  we  have 

^solution  —  *H2O  =  K> 

and 

Av  =  Aoo  =  1000  K.V. 
Hence 

=  HXXU' 

or  if  m  denotes  the  concentration  in  gram-equivalents  per  liter, 
we  have 

1       1000 K 


Thus,  Bottger  found  for  a  saturated  solution  of  silver  chloride  at 
20°,  K'  =  1.374  X  10~6.  Deducting  the  specific  conductance  of 
the  water  at  this  temperature,  we  have 

K  =  1.374  X  10-6  -  0.044  X  10~6  =  1.33  X  lO"6. 

Since  the  value  of  A»,  at  20°,  for  silver  chloride,  determined  from 
the  table  of  ionic  conductances,  is  125.5,  we  have 

TO  =  1000XL33Xlfr-»  =  j  Q6  x  ]0_5  gr  _equiv  AgC1  per  nter. 


416 


THEORETICAL  CHEMISTRY 


Temperature  Coefficient  of  Conductance.  When  the  temper- 
ature of  a  solution  of  an  electrolyte  is  raised,  the  equivalent  con- 
ductance usually  increases.  The  increase  in  conductance  is  due, 
not  to  an  increase  in  the  ionization,  but  to  the  greater  velocity 
of  the  ions  caused  by  the  diminution  of  the  viscosity  of  the  solution. 
According  to  Kohlrausch,  the  relation  between  conductance  and 
temperature  may  be  approximately  expressed  by  the  following 
equation, 

A|=A18o}l  +0($ 


where  /3  is  the  temperature  coefficient,  or  change  in  conductance 
for  1°  C.     Solving  the  equation  for  /3,  we  have 

A,  -  A180 


A     a       iz\ 

•i*-18°(l   —    io) 

The  temperature  coefficients  of  several  of  the  more  common  elec- 
trolytes are  given  in  the  accompanying  table. 


TEMPERATURE  COEFFICIENTS  OF  CONDUCTANCE. 


Electrolyte. 

Temperature 
Coefficient. 

Nitric  acid.  . 

0  0163 

Sulphuric  acid 

0  0164 

Hydrochloric  acid  
Potassium  hydroxide  
Potassium  nitrate 

0.0165 
0.0190 
0  0211 

Potassium  iodide  . 

0  0212 

Potassium  bromide  .... 

0  0216 

Potassium  chlorate.  .  . 

0.0216 

Silver  nitrate  

0  0216 

Potassium  chloride  .  .  . 

0.0217 

Ammonium  chloride  
Potassium  sulphate  
Copper  sulphate 

0.0219 
0.0223 
0  0225 

Sodium  chloride  
Sodium  sulphate.  . 

0.0226 
0  0234 

Zinc  sulphate  .  . 

0  0250 

The  temperature  coefficient  of  conductance  is  not,  however,  a 
simple  linear  function  of  the  temperature.     The  following  empiri- 


ELECTRICAL   CONDUCTANCE  417 

cal  equations,  expressing  equivalent  conductance  at  infinite  dilu- 
tion at  any  temperature  t  in  terms  of  the  conductance  at  18°,  have 
been  derived  by  Kohlrausch:  — 

A**  =  Aooiso  { 1  +  a  (t  -  18)  +  ft  (t  -  18)2}, 
and 

ft  =  0.0163  (a  -  0.0174). 

When  the  values  of  Aooi8o,  a,  and  ft,  as  determined  for  a  large 
number  of  electrolytes,  are  substituted  in  the  above  equation,  he 
showed  that  Aoo<  becomes  equal  to  zero  at  a  temperature  approx- 
imating to  —  40°.  Kohlrausch  suggested  that  each  ion  moving 
through  the  solution  carries  with  it  an  " atmosphere"  of  solvent, 
and  that  the  resistance  offered  to  the  motion  of  the  ion  is  simply 
the  frictional  resistance  between  masses  of  pure  water.  This 
view  is  in  harmony  with  the  solvate  theory  discussed  in  an  earlier 
chapter.  Washburn  *  has  calculated  the  degree  of  ionic  hydration 
for  several  ions.  He  finds,  for  example,  that  the  hydrogen  ion 
carries  with  it  0.3  molecule  of  water,  while  the  lithium  ion  is 
hydrated  to  the  extent  of  4.7  molecules  of  water. 

Conductance  at  High  Temperatures  and  Pressures.  The 
conductance  of  several  typical  electrolytes,  at  temperatures  rang- 
ing from  that  of  the  room  up  to  306°,  have  been  measured  by  A.  A. 
Noyes  and  his  co-workers,  f  These  determinations  were  made  in 
a  conductance  cell  especially  constructed  to  withstand  high 
pressures. 

The  results  show  that  the  values  of  A<x>  for  binary  electrolytes 
become  more  nearly  equal  with  rise  of  temperature.  This  may 
be  taken  as  an  indication  of  the  fact  that  the  ionic  velocities  tend 
to  become  more  nearly  equal  as  the  temperature  rises.  The 
conductance  of  ternary  electrolytes  increases  uniformly  with  the 
temperature,  and  attains  values  which  are  considerably  greater 
than  those  reached  by  binary  electrolytes.  This  is  what  might 
be  expected,  since  if  an  ion  is  bivalent,  as  in  a  ternary  electrolyte, 
the  driving  force  is  greater,  and  the  ion  must  move  faster,  and, 
consequently,  the  conductance  must  be  greater. 

*  Jour.  Am.  Chem.  Soc.,  30,  322  (1909). 

t  Publication  of  Carnegie  Institution,  No.  63. 


418  THEORETICAL  CHEMISTRY 

The  temperature  coefficient  of  conductance  for  binary  elec- 
trolytes is  greater  between  100°  and  156°,  than  below  or  above 
these  temperatures.  The  temperature  coefficients  of  ternary 
electrolytes  increases  uniformly  with  rising  temperature.  In  the 
case  of  acids  and  bases,  the  rate  of  increase  in  conductance  steadily 
diminishes  as  the  temperature  rises.  The  ionization  decreases 
regularly  with  rise  in  temperature,  the  temperature  coefficient  of 
ionization  being  small  between  18°  and  100°.  The  effect  of  pres- 
sure on  conductance  was  studied  by  Fanjung.*  He  found  that 
the  conductance  increases  slightly  with  increasing  pressure. 
This  result  he  interprets  as  being  due  to  increased  ionic  velocity 
rather  than  to  an  increase  in  the  number  of  ions  present  in  the 
solution. 

Conductance  of  Non-aqueous  Solutions.  A  large  amount  of 
interesting  and  important  work  has  been  done  in  recent  years 
upon  the  electrical  conductance  of  solutions  in  non-aqueous  sol- 
vents. 

It  is  impossible  to  give  even  a  brief  survey  of  the  results  of  these 
investigations,  and  we  must  limit  ourselves  to  the  statement  of 
the  following  general  conclusions :  — f 

(1)  The  conditions  in  non-aqueous  solutions  are  much  more 
complex  than  in  aqueous  solutions. 

(2)  In  general,  the  laws  which  have  been  found  to  apply  to  aque- 
ous solutions  also  apply  to  non-aqueous  solutions. 

(3)  Different   solvents  appear  to   have  different  dissociating 
powers. 

(4)  The  dissociating  power  appears  to  run  parallel  with  the 
dielectric  constant  of  the  solvent. 

Many  interesting  phenomena  present  themselves  in  connec- 
tion with  the  conductance  of  electrolytes  in  mixed  solvents,  but 
for  an  account  of  this  work  the  student  must  consult  the  original 
papers  of  Jones  and  his  students,  f 

*  Zeit.  phys.  Chem.,  14,  673  (1894). 

t  " Elektrochemie  der  nichtwassrigen  Losungen,"  by  G.  Carrara,  Ahren'a 
"Sammlung  Chemischer  und  chemisch-technischer  Vortraege,"  Vol.  XII. 
J  Publication  of  the  Carnegie  Institution,  No.  80. 


ELECTRICAL   CONDUCTANCE 


419 


Ionizing  Power  of  Solvents.  Thomson  *  and  Nernst  f  pointed 
out  that  if  the  forces  which  hold  the  atoms  in  the  molecule  are  of 
electrical  origin,  then  those  liquids  which  possess  large  dielectric 
constants  should  have  correspondingly  great  ionizing  power. 
This  is  a  direct  consequence  of  Coulomb's  law  of  electrostatic 
attraction,  which  may  be  expressed  by  the  equation, 


/  = 


Kd? 


in  which  51  and  #2  denote  two  electric  charges,  d  the  distance 
between  them,  /  the  force  of  attraction  and  K  the  dielectric  con- 
stant. Obviously  the  larger  K  becomes,  the  smaller  will  be  the 
value  of/;  i.  e.,  the  more  likely  the  molecule  will  be  to  break  down 
into  ions.  That  the  above  relation  is  approximately  true  may  be 
seen  from  the  following  table :  — 

DIELECTRIC  CONSTANTS. 


Solvent. 

K 

Ionizing  Power. 

Benzene  

2.3 

Extremely  weak 

Ethyl  ether  

4.1 

Weak 

Ethyl  alcohol  

25 

Fairly  strong 

Formic  acid  

62 

Strong 

Water 

80 

Very  strong 

Hydrocyanic  acid 

96 

Very  strong 

Dutoit  and  Aston  J  have  suggested  that  there  is  a  connection 
between  the  ionizing  power  of  a  solvent  and  its  degree  of  associa- 
tion, and  Dutoit  and  Friderich  §  conclude  that  the  values  of  A«, 
for  a  given  electrolyte  dissolved  in  different  solvents,  are  a  direct 
function  of  the  degree  of  association  and  an  inverse  function  of 
the  viscosity  of  the  solvents.  Water  and  the  alcohols  furnish 
good  illustrations  of  the  truth  of  this  generalization. 

*  Phil.  Mag.,  36,  320  (1893). 

t  Zeit.  phys.  Chem.,  13,  531  (1894). 

t  Compt.  rend.,  125,  240  (1897). 

§  Bull.  Soc.  Chim.  [3],  19,  321  (1898). 


420 


THEORETICAL  CHEMISTRY 


Conductance  of  Fused  Salts.  While  solid  salts  are  exceed- 
ingly poor  conductors  of  electricity,  yet  as  the  temperature  is 
raised  their  conductance  increases  until  at  their  melting  point 
they  may  be  grouped  with  good  conductors.  There  is  no  sudden 
increase  in  conductance  at  the  melting  point.  The  specific 
conductance  of  a  fused  salt  may  exceed  the  specific  conductance 
of  the  most  concentrated  aqueous  solutions,  but  owing  to  the  high 
concentration  the  equivalent  conductance  is  much  less.  The 
following  table  gives  the  specific  and  equivalent  conductance  of 
fused  silver  nitrate :  — 


Temperature, 
degrees. 

Sp.  Cond. 

Equiv.  Cond. 

218  (melt,  pt.) 

0.681 

29.2 

250 

0.834 

36.1 

300 

1.049 

46.2 

350 

1.245 

55.4 

The  specific  conductance  of  a  60  per  cent  aqueous  solution  of 
silver  nitrate  at  18°  is  0.208  reciprocal  ohms. 

If  the  salts  are  impure  the  conductance  is  raised,  the  effect  of 
impurities  being  apparent  even  before  the  salts  have  reached  their 
melting  points.  This  is  analogous  to  the  behavior  of  solutions, 
and  suggests  that  the  impurity  functions  in  the  salt  mixture  as  a 
dissolved  solute.* 

PROBLEMS. 

1.  An  aqueous  solution  of  copper  sulphate  is  electrolyzed  between 
copper  electrodes  until  0.2294  gram  of  copper  is  deposited.     Before  elec- 
trolysis the  solution  at  the  anode  contained  1.1950  grams  of  copper,  after 
electrolysis  1.3600  grams.     Calculate  the  transport  numbers  of  the  two 
ions,  Cu"  and  S04".  Ans.  n  =  0.28,  1  -  n  =  0.72. 

2.  A  solution  containing  0.1605  per  cent  of  NaOH  was  electrolyzed 
between   platinum   electrodes.     After   electrolysis    55.25   grams   of   the 
cathode  solution  contained  0.09473  gram  of  NaOH,  whilst  the  concen- 
tration of  the  middle  portion  of  the  electrolyte  was  unchanged.     In  a 

*  For  a  complete  treatment  of  fused  electrolytes  the  student  is  advised  to 
consult,  "Die  Elektrolyse  geschmolzener  Salze, "  by  Richard  Lorenz. 


ELECTRICAL   CONDUCTANCE  421 

silver  coulometer  the  equivalent  of  0.0290  gram  of  NaOH  was  deposited 
during  electrolysis.  Calculate  the  transport  numbers  of  the  Na"  and  OH' 
ions.  Am.  n  =  0.791,  1  -  n  =  0.209. 

3.  In  a  0.01  molar  solution  of  potassium  nitrate,  the  transport  num- 
bers of  the  cation  and  anion  are,  respectively,  0.503  and  0.497.     Find  the 
equivalent  conductances  of  the  two  ions  in  this  solution  having  given  that 
its  specific  conductance  is  0.001044.  Am.   lc  =  52.5,  la  =  51.9. 

4.  The  absolute  velocity  of  the  Ag*  ion  is  0.00057  cm.  per  sec.,  and  that 
of  the  Cl'  ion  is  0.00059  cm.  per  sec.      Calculate  the  equivalent  con- 
ductance of  an  infinitely  dilute  solution  of  silver  chloride. 

5.  The  equivalent  conductance  of  an  infinitely  dilute  solution  of  am- 
monium chloride  is  130;  the  ionic  conductances  of  the  ions  OH'  and  Cl' 
are  174  and  65.44  respectively.     Calculate  the  equivalent  conductance 
of  ammonium  hydroxide  at  infinite  dilution.  Am.   A«  =  238.56. 

6.  The  equivalent  conductance  of  a  molar  solution  of  sodium  nitrate 
at  18°  is  66;    its  conductance  at  infinite  dilution  is  105.3.     What  is  the 
degree  of  ionization  in  the  molar  solution?        Ans.  a  =  62.6  per  cent. 

7.  The  specific  conductance  of  a  saturated  solution  of  AgCN  at  20° 
is  1.79  X  10~6  and  the  specific  conductance  of  water  at  the  same  temper- 
ature is  0.044  X  10~6  reciprocal  ohms.     The  equivalent  conductance  at 
infinite  dilution  is  115.5.     Calculate  the  solubility  of  AgCN  in  grams  per 
liter.  Ans.   2.02  X  10~3  gram/liter. 

8.  The  equivalent  conductance  at  18°  of  a  solution  of  sodium  sulphate 
containing  0.1  gram-equivalent  of  salt  per  liter  is  78.4,  the  conductance 
at  infinite  dilution  is  113  reciprocal  ohms.     What  is  the  value  of  i  for 
the  solution?    What  is  its  osmotic  pressure? 

Ans.   i  =  2.388;  osmotic  pressure  =  2.85  atmos. 

9.  The  freezing-point  of  a  0.1  molar  solution  of  CaCl2  is  —  0°.482. 
(a)  Calculate  the  degree  of   ionization  (freezing  point  constant  =  1.89 
for  one  mol  per  liter),     (b)  Calculate  the  degree  of  ionization  from  the 
equivalent  conductance  at  18°,  which  is  82.79  reciprocal  ohms,  whilst  the 
equivalent  conductance  of  CaCU  at  infinite  dilution  is  115.8  reciprocal 
ohms.  Ans.   (a)  a  =  0.774;    (6)  a  =  0.715. 


CHAPTER  XIX. 
ELECTROLYTIC  EQUILIBRIUM   AND   HYDROLYSIS. 

Ostwald's  Dilution  Law.  It  has  been  shown  in  preceding 
chapters  that  the  law  of  mass  action  is  applicable  to  chemical 
equilibria  in  both  gaseous  and  liquid  systems.  We  now  proceed 
to  show  that  it  applies  equally  to  electrolytic  equilibria.  When 
acetic  acid  is  dissolved  in  water  it  dissociates  according  to  the 
equation 

CHsCOOH  <=±  CH3COO'  +  H*. 

Let  one  mol  of  acetic  acid  be  dissolved  in  water  and  the  solution 
diluted  to  v  liters,  and  let  a  denote  the  degree  of  dissociation. 

Then,  the  concentration  of  the  undissociated  acid  is  -     —  ,  and 

the  concentration  of  the  ions  is--     Applying  the  law  of  mass 
action,  we  have 


or 


where  K  is  the  equilibrium  or  ionization  constant. 

This  equation  expressing  the  relation  between  the  degree  of 
ionization  and  dilution,  was  derived  by  Ostwald*  and  is  known 

as  the  Ostwald  dilution  law.     Since  a  =  —  -  -  ,  we  may  substitute 

Aoo 

this  value  of  a  in  equation  (1)  and  obtain  the  expression 


oo  (Aw  —  Ar)  v 


=  K.  (2) 


*  Zeit.  phys.  Chem.,  2,  36  (1888);  3,  170  (1889). 
422 


ELECTROLYTIC  EQUILIBRIUM  AND  HYDROLYSIS       423 


The  dilution  law  may  be  tested  by  substituting  the  value  of  a, 
corresponding  to  any  dilution  v,  in  the  equation  and  calculating 
the  value  of  the  ionization  constant,  K\  the  value  of  a  at  any 
other  dilution  may  then  be  calculated  and  compared  with  the 
value  determined_by  direct  experiment.  The  following  table  gives 
the  results  obtained  with  acetic  acid  at  14°.l,  K  being  equal'  to 
0.0000178:  — 


»  (in  liters). 

aX102  (calc.). 

aX102  (obs.). 

0.994 

0.42 

0.40 

2.02 

0.60 

0.614 

15.9 

1.67 

1.66 

18.1 

1.78 

1.78 

1,500.0 

15.0 

14.7 

3,010.0 

20.2 

20.5 

7,480.0 

30.5 

30.1 

15,000.0 

40.1 

40.8 

As  will  be  seen,  the  agreement  between  the  observed  and  cal- 
culated values  is  very  close.  The  table  also  shows  to  how  small 
an  extent  the  molecules  of  acetic  acid  are  broken  down  into  ions, 
a  molar  solution  being  dissociated  less  than  0.5  per  cent.  The 
dilution  law  holds  for  nearly  all  organic  acids  and  bases,  but  fails 
to  apply  to  salts,  strong  acids,  and  strong  bases.  When  a  is 
small,  the  term  (1  —  a)  does  not  differ  appreciably  from  unity, 
and  equation  (1)  becomes 


or 


(3) 


On  the  other  hand,  when  a  cannot  be  neglected,  we  have,  on 
solving  equation  (2)  for  a, 

V  J\.         ,        A      I         T^        .        C/    J.*. 

The    method    of   derivation   indicates   that   the   dilution   law   is 
only  strictly   applicable    to    binary  electrolytes   and,  therefore, 


424  THEORETICAL  CHEMISTRY 

it  is  improbable  that  it  will  hold  for  electrolytes  yielding  more 
than  two  ions.  It  has  been  found,  however,  that  organic  acids 
whether  they  are  mono-,  di-,  or  polybasic  always  ionize  as 
a  monobasic  acid  up  to  the  dilution  at  which  a  =  50  per 
cent.  This  means  that  the  dilution  law  is  applicable  to  poly- 
basic  acids  up  to  that  dilution  at  which  the  acid  is  50  per  cent 
ionized. 

Strength  of  Acids  and  Bases.  There  are  several  methods  by 
which  the  relative  strengths  of  acids  can  be  estimated.  A  method 
which  has  proved  of  great  value  is  that  in  which  two  different 
acids  are  allowed  to  compete  for  a  certain  base,  the  amount  of 
which  is  insufficient  to  saturate  both  of  them.  Suppose  equiva- 
lent weights  of  nitric  and  dichloracetic  acids  together  with  sufficient 
potassium  hydroxide  to  saturate  one  acid  completely  are  taken: 
we  then  determine  the  position  of  the  equilibrium  represented  by 
the  equation 

HN03  +  CHC12  •  COOK  <=±  CHC12  •  COOH  +  KNO3. 

In  order  to  determine  the  conditions  of  equilibrium  we  may  make 
use  of  any  method  which  does  not  disturb  this  equilibrium.  Since 
ordinary  chemical  methods  are  excluded  on  this  account,  we 
employ  any  physical  property  which  is  capable  of  exact  measure- 
ment and  differs  sufficiently  in  the  two  systems,  as  for  example, 
the  change  in  volume,  or  the  thermal  change,  accompanying 
neutralization.  Thus,  Ostwald  *  found  that  when  one  mol  of 
potassium  hydroxide  is  neutralized  by  nitric  acid  in  dilute  solu- 
tion, the  volume  increases  approximately  20  cc.  When  one  mol 
of  potassium  hydroxide  is  neutralized  by  dichloracetic  acid,  how- 
ever, the  increase  in  volume  is  13  cc.  It  is  evident,  therefore,  that 
if  nitric  acid  completely  displaces  dichloracetic  acid  as  represented 
by  the  above  equation,  the  increase  in  volume  will  be  20  —  13 
=  7  cc.;  if  no  displacement  occurs,  then  the  volume  will  remain 
constant.  He  found  that  the  volume  actually  increased  5.67  cc. 
Therefore,  the  reaction  represented  by  the  upper  arrow  has  pro- 
ceeded to  the  extent  of  5.67  -^  7  =  80  per  cent.  That  is  to  say, 

*  Jour,  prakt.  Chem.  [2],  18,  328  (1878). 


ELECTROLYTIC  EQUILIBRIUM   AND  HYDROLYSIS       425 


in  the  competition  of  the  two  acids  for  the  base,  the  nitric  acid 
has  taken  80  per  cent  and  the  dichloracetic  acid  has  taken  20  per 
cent,  or  the  relative  strengths  of  the  two  acids  are  in  the  ratio  of 
80  :  20,  or  4  :  1. 

The  relative  strengths  of  acids  can  also  be  determined  from  their 
catalytic  effect  on  the  rates  of  certain  reactions,  such  as  the 
hydrolysis  of  esters  or  the  inversion  of  cane  sugar. 

The  order  of  the  activity  of  acids  is  the  same  whether  measured 
by  equilibrium  or  kinetic  methods.  Arrhenius  pointed  out  that 
the  relative  strengths  of  acids  can  be  readily  determined  from 
their  electrical  conductance.  The  order  of  the  strengths  of  acids 
as  determined  by  equilibrium  and  kinetic  methods  is  the  same  as 
that  of  their  electrical  conductances  in  equivalent  solutions. 
This  is  well  illustrated  by  the  following  table  in  which  the  three 
methods  are  compared,  hydrochloric  acid  being  taken  as  the 
standard  of  comparison:  — 


Acid. 

Method  Employed. 

Equilibrium. 

Kinetic. 

Conductance. 

HC1.. 

100 
100 

49 
9 

100 
100 
53.6 
4.8 
0.4 

100 
99.6 
65.1 
4.8 
1.4 

HNO3 

H2SO4... 

CH2C1COOH 

CHsCOOH. 

The  results  of  these  and  other  experiments  warrant  the  con- 
clusion that  the  strength  of  an  acid  is  determined  by  the  number 
of  hydrogen  ions  which  it  yields.  It  is  important  to  note  that  the 
electrical  conductance  of  an  acid  is  not  directly  proportional  to 
its  hydrogen  ion  concentration;  the  relatively  high  velocity  of 
the  H  ion  is  the  cause  of  the  approximate  proportionality  between 
these  two  variables.  In  the  case  of  a  weak  acid,  the  value  of  the 
ionization  constant  may  be  taken  as  a  measure  of  the  strength  of 
the  acid.  The  following  table  gives  the  values  of  the  ionization 
constants  at  25°  for  several  different  acids. 


426 


THEORETICAL  CHEMISTRY 
IONIZATION  CONSTANTS  OF  ACIDS. 


Acid. 

lonization 
Constant. 

Acetic  acid 

0  0000180 

Monochloracetic  acid  .... 

0  00155 

Trichloracetic  acid  .  . 

1  21 

Cyanacetic  acid  
Formic  acid  

0.0037 
0.000214 

Carbonic  acid  

3040  XlO-10 

Hydrocyanic  acid  

570  XlO-10 

Hydrogen  sulphide  

13  XlO-10 

Phenol  

1.3  XlO-10 

Since  for  a  weak  acid,  a  =  \^vK,  it  follows  that  for  two  weak 
acids  at  the  same  dilution,  we  may  write 


01 

0:2 

or  the  ratio  of  the  degrees  of  ionization  of  the  two  acids  is  equal  to 
the  square  root  of  the  ratio  of  their  ionization  constants.  Thus, 
from  the  data  given  in  the  foregoing  table  for  acetic  and  mono- 

chloracetic  acids,  we  have      

'0.000018       1 


ai  =    /0.0 
a,      V  0.( 


a2       *    u.00155       9.3 
or  the  effect  of  replacing  one  atom  of  hydrogen  in  the  methyl 
group  of  acetic  acid  increases  the  strength  of  the  acid  about  nine 
times. 

Just  as  the  hydrogen  ion  concentration  of  acids  determines  their 
strength,  so  the  strength  of  bases  is  determined  by  the  concen- 
tration of  hydroxyl  ions.  The  strength  of  bases  may  be  estimated 
by  methods  similar  to  those  employed  in  determining  the  strength 
of  acids.  Thus,  two  different  bases  may  be  allowed  to  compete 
for  an  amount  of  acid  sufficient  to  saturate  only  one  of  them;  or 
a  catalytic  method  developed  by  Koelichen  *  may  be  used.  This 
method  is  based  upon  the  effect  of  hydroxyl  ions  on  the  rate  of 
condensation  of  acetone  to  diacetonyl  alcohol,  as  represented  by 
the  equation 

2CH3COCH3  =  CH3COCH2C  (CH3)2OH. 
*  Zeit.  phys.  Chem.,  33,  129  (1900). 


ELECTROLYTIC  EQUILIBRIUM   AND   HYDROLYSIS       427 

In  addition  to  these  two  methods,  the  method  of  electrical  con- 
ductance is  also  applicable.  The  agreement  between  the  results 
obtained  by  the  three  methods  is  quite  satisfactory.  The  alkali 
and  alkaline  earth  hydroxides  are  very  strong  bases  and  are  dis- 
sociated to  about  the  same  extent  as  equivalent  solutions  of 
hydrochloric  and  nitric  acids,  while  on  the  other  hand,  ammonia 
and  many  of  the  organic  bases  are  very  weak.  The  following 
table  gives  the  ionization  constants  of  several  typical  bases:  — 


IONIZATION   CONSTANTS  OF  BASES. 


Base. 

Ionization 
Constant. 

Ammonia.         .    ... 

0  000023 

Methylamine  ... 

0  00050 

Trimethylamine  

0  000074 

Pyridine  

2.5X10-10 

Aniline  

1  .  1  X  10-10 

Mixtures  of  Two  Electrolytes  with  a  Common  Ion.  Just  as 
the  dissociation  of  a  gaseous  substance  is  diminished  by  the  addi- 
tion of  an  excess  of  one  of  the  products  of  dissociation,  so  the 
ionization  of  weak  acids  and  bases  is  depressed  by  the  addition  of 
a  salt  with  an  ion  common  to  the  acid  or  the  base.  If  the  degree 
of  ionization  of  a  salt  with  an  ion  in  common  with  an  acid,  or  a 
base  is  represented  by  a',  and  n  denotes  the  number  of  molecules 
of  salt  present,  then  the  equation  of  equilibrium  of  the  acid  or 
base  will  be 

(not  +  a)a  =  Kv(l-  a), 

where  a  is  the  degree  of  ionization  of  the  acid  or  base.  For  very 
weak  acids  and  bases,  a  is  so  small  that  1  —  a  does  not  differ 
appreciably  from  unity,  and  since  a  is  practically  independent  of 
the  dilution,  we  obtain 

no.  =  Kv 
or 

Kv 

a  =  — • 
n 


428  THEORETICAL  CHEMISTRY 

That  is,  the  ionization  of  a  weak  acid  or  base,  in  the  presence  of 
one  of  its  salts,  is  approximately  inversely  proportional  to  the 
amount  of  salt  present. 

In  many  of  the  processes  of  analytical  chemistry,  advantage  is 
taken  of  the  action  of  neutral  salts  on  the  ionization  of  weak  acids 
and  bases.  Thus,  while  the  concentration  of  hydroxyl  ions  in 
ammonium  hydroxide  is  sufficient  to  precipitate  magnesium  hy- 
droxide from  solutions  of  magnesium  salts,  the  presence  of  a  small 
amount  of  ammonium  chloride  depresses  the  ionization  of  the 
ammonium  hydroxide  to  such  an  extent  that  precipitation  no 
longer  takes  place. 

Isohydric  Solutions.  Arrhenius  *  was  the  first  to  point  out 
what  relation  must  exist  between  solutions  of  two  electrolytes 
with  a  common  ion,  in  order  that,  when  mixed  in  any  proportions, 
they  may  not  exert  any  mutual  influence.  He  showed  that  when 
the  concentration  of  the  common  ion  in  each  of  the  two  solutions 
is  the  same  before  mixing,  no  alteration  in  the  degree  of  ionization 
will  occur  after  mixing.  Such  solutions  are  said  to  be  isohydric. 
Thus,  an  aqueous  solution  containing  one  mol  of  acetic  acid  in 
8  liters,  is  isohydric  with  an  aqueous  solution  containing  one  mol 
of  hydrochloric  acid  in  667  liters.  On  mixing  these  two  solutions 
the  hydrogen  ion  concentration  remains  unchanged,  and  if  the 
mixture  is  treated  with  a  small  amount  of  sodium  hydroxide, 
equal  amounts  of  sodium  acetate  and  sodium  chloride  will  be 
formed. 

That  isohydric  solutions  may  be  mixed  without  altering  their 
respective  ionizations  may  be  shown  in  the  following  manner :  — 
Let  C  and  c  denote  the  concentrations  of  the  undissociated  por- 
tions, and  CA,  Cz,  CA,  and  c2  denote  the  concentrations  of  the  dis- 
sociated portions  of  two  electrolytes,  and  let  C%  and  Cz  correspond 
to  two  different  ions. 

Then,  we  have 

kc  =  cACv,  (1) 

and 

KC  -  CAC*.  (2) 

*  Wied.  Ann.,  30,  51  (188,7). 


ELECTROLYTIC  EQUILIBRIUM  AND  HYDROLYSIS       429 

If  v  liters  of  the  first  solution  be  mixed  with  V  liters  of  the  second 
solution,  the  concentrations  of  the  undissociated  portions,  and  of 
the  dissimilar  ions,  will  be 

Cv  cv  C2V  w 

while  the  concentration  of  the  common  ion  A,  becomes 

CAV  +  cAv 

Applying  the  law  of  mass  action,  we  have 
j         CA.V  +  CA.V 

KC   =    pr— C2,  (3) 

and 

"W /~i  ^-/  A.  '        T~   ^ 'Ay   s~i  t  j\ 

KC  v  +  v     C2  •  (4) 

But  equations  (3)  and  (4)  only  become  identical  with  equations  (1) 
and  (2)  when  CA  =  CA,  or  in  other  words,  no  change  in  the  degree 
of  dissociation  takes  place  after  the  two  solutions  are  mixed. 

lonization  of  Strong  Electrolytes.  It  has  already  been  men- 
tioned that  the  Ostwald  dilution  law,  which  is  a  direct  conse- 
quence of  the  law  of  mass  action,  applies  only  to  weak  electrolytes. 
Just  why  the  law  of  mass  action  should  fail  to  apply  to  strong 
electrolytes  is  not  known,  but  several  possible  causes  have  been 
suggested  to  account  for  its  failure.  One  of  the  most  plausible 
explanations  is  that  advanced  by  Biltz,*  who  attributes  the 
failure  of  the  law  of  mass  action  when  applied  to  strong  electrolytes, 
to  hydration  of  the  solute.  If  the  ions  become  associated  with 
a  large  proportion  of  the  solvent,  the  effective  ionic  concentration 
would  then  be  the  ratio  of  the  amount  of  the  ion  present  to  that 
of  the  free  solvent,  instead  of  to  the  total  solvent,  as  ordinarily 
calculated.  This  view  is  in  harmony  with  certain  facts  which 
have  been  adduced  in  favor  of  the  theory  of  solvation.  While 
the  Ostwald  dilution  law  does  not  apply  to  strongly  ionized  elec- 
trolytes, certain  empirical  expressions  have  been  derived  which 
*  Zeit.  phys.  Chem.,  40,  218  (1902). 


430 


THEORETICAL  CHEMISTRY 


hold  fairly  well  over  a  wide  range  of  dilution.     Thus,  Rudolph!  * 
showed  that  the  equation 


(1  -  a)  Vv 

gives  approximately  constant  values  for  K'  for  strong  electrolytes. 
The  following  table  gives  the  results  obtained  with  solutions  of 
silver  nitrate  at  25°;  the  numbers  in  the  third  column  being  cal- 
culated by  means  of  the  Ostwald  dilution  law,  while  those  in  the 
fourth  column  are  calculated  by  means  of  Rudolphi's  dilution  law. 


» 

a 

K 

K' 

16 

0.8283 

0.253 

.11 

32 

0.8748 

0.191 

.16 

64 

0.8993 

0.127 

.06 

128 

0.9262 

0.122 

.07 

256 

0.9467 

0.124 

.08 

512 

0.9619 

0.125 

.09 

The  Rudolphi  equation  was  modified  by  van't  Hoff  f  to  the 
form 


/I    \2 

This  equation  holds  even  more  closely  than  that  of  Rudolphi. 
Of  the  more  recent  empirical  equations  which  have  been  derivec 
to  express  the  change  of  conductance  of  an  electrolyte  with  dilu- 
tion, the  equations  of  Kraus  *  and  Bates  f  deserve  special  men- 
tion. 
The  equation  of  Kraus  has  the  following  form:  — 

/A         \  9  /~^ 

I  A??  V          C  ,    , , 


\A0?7o/ 


(i  -  ^) 


In  this  equation  A  is  the  conductance  of  the  electrolyte  whose  con- 
centration C  is  expressed  in  mols  per  liter,  77/770  is  the  ratio  of  the 
viscosity  of  the  solution  to  that  of  the  solvent,  and  k,  k',  A,  and  A0 

*  Jour.  Am.  Chem.  Soc.,  35,  1412  (1913). 
t  Ibid.,  37,  1421  (1915). 


ELECTROLYTIC  EQUILIBRIUM  AND  HYDROLYSIS      431 


are  empirical  constants  the  values  of  which  are  so  chosen  as  to 
insure  close  agreement  between  the  observed  and  calculated  values 
of  the  conductance. 

The  equation  of  Bates  is  similar  to  that  of  the  preceding  equa- 
tion, except  that  the  logarithm  of  the  left-hand  side  of  the  equation 
is  substituted  for  the  original  expression  of  Kraus.  The  equation 
of  Bates  takes  the  form :  — 

£ =  k  +  k>(c±f\h. 

Ar?\  V 


i         /  lvri 

lOglO  (-T 


The  constants  k,  k',  and  h  are  purely  empirical  as  in  the  equation 
of  Kraus,  but  A0  denotes  the  equivalent  conductance  at  infinite 
dilution. 

A  comparison  between  the  two  equations  is  afforded  by  the  fol- 
lowing table  in  which  is  recorded  the  observed  and  calculated 
values  of  the  "corrected  "  equivalent  conductance,  A^/r/o,  for 
solutions  of  potassium  chloride  at  18°. 

COMPARISON  OF  THE  EQUATIONS  OF  KRAUS  AND  BATES. 


c 

•n/no 

A  (obs.) 

AIJ/IJO 

A,/%  (K.) 

A.,/,,,  (B.) 

3 

0.9954 

88.3 

87.89 

87.4 

89.3 

2 

0.9805 

92.53 

90.73 

90.9 

91.9 

1 

0.982 

98.22 

96.5 

96.4 

96.53 

0.5 

0.9898 

102.36 

101.32 

101.1 

101.29 

0.2 

0.9959 

107.90 

107.46 

107.6 

107.43 

0.1 

0.9982 

111.97 

111.77 

1U.  9 

111.73 

0.05 

0.9991 

115.69 

115.59 

115.5 

115.58 

0.02 

0.9996 

119.90 

119.85 

119.8 

119.83 

0.01 

0.9998 

122.37 

122.35 

122.4 

122.32 

0.005 

0.9999 

124.34 

124.33 

124.4 

124.38 

0.002 

1.0000 

126.24 

126.24 

126.3 

126.31 

0.001 

127.27 

127.27 

127.2 

127.32 

0.0005 

128.04 

128.04 

127.6 

128.05 

0.0002 

128.70 

128.70 

127.9 

.128.68 

0.0001 

129.00 

129.00 

128.1 

128.96 

0.0 

129  50 

129.50 

128.3 

129.50 

It  will  be  observed  that  for  dilute  solutions,  the  ratio  77/170  is 
practically  unity  and,  furthermore,  that  the  value  of  (7A/A0  is  so 
small  that  the  second  term  on  the  right-hand  side  of  both  equations 
is  negligible  in  comparison  with  the  value  of  k. 


432 


THEORETICAL  CHEMISTRY 


Therefore,  under  these  conditions,  both  equations  reduce  to  the 
form 

A2C 
— TT TT  =  constant, 

Ao  (A0  —  A) 

which  will  be  recognized  as  identical  with  Ostwald's  dilution  law 
as  given  on  page  422. 

Heat  of  lonization.  The  heat  of  ionization  of  an  electrolyte 
can  be  calculated  by  means  of  the  reaction  isochore  equation  of 
van't  Hoff  (see  p.  324),  provided  the  degree  of  ionization  at  two 
different  temperatures  is  known. 


Since 
and 


l  — 


(1  - 


K  a*Z 

(1  -  «2)  v' 

it  follows  that  the  heat  of  ionization  may  be  calculated  by  means 
of  the  equation 


2.306  R     log 


,, 

(1  ~ 


-  log  ,, 

V          (1 


Arrhenius  *  has  shown  that  this  equation  also  applies  to  those 
electrolytes  which  do  not  obey  the  Ostwald  dilution  law.  Some 
of  the  results  obtained  by  Arrhenius  are  given  in  the  accompany- 
ing table  :  — 


Electrolyte. 

Temperature. 

Calories. 

Acetic  acid                                                                  \ 

35° 

386 

Propionic  acid                                                            \ 

21°.5 
35° 

-28 
557 

Butyric  acid  ] 

21°.5 
35° 

183 
935 

Phosphoric  acid  < 

21°.  5 
35° 

427 

2458 

Hydrochloric  acid  

21°.  5 
35° 

2103 
1080 

Potassium  chloride   .  . 

35° 

362 

Potassium  bromide  

35° 

425 

Potassium  iodide  

35° 

916 

Sodium  chloride  

35° 

454 

Sodium  hydroxide 

35° 

1292 

Sodium  acetate  

35° 

391 

Zeit.  phys.  Chem.,  4,  96  (1889). 


ELECTROLYTIC  EQUILIBRIUM   AND  HYDROLYSIS       433 

It  will  be  found  that  the  values  of  the  heats  of  ionization  given 
in  this  table  do  not  agree  with  the  values  calculated  for  these 
same  substances  from  the  data  given  in  the  table  on  page  307. 
The  reason  for  this  lack  of  agreement  is,  that  the  data  of  the  earlier 
table  refer  to  the  heat  of  formation  of  the  ions  from  the  dissolved 
substance,  whereas  the  data  of  the  table  just  given  represent  the 
combined  thermal  effects  of  solution  and  ionization. 

The  Solubility  Product.  While  the  law  of  mass  action  does 
not  in  general  apply  to  the  equilibrium  between  the  dissociated 
and  undissociated  portions  of  an  electrolyte,  —  except  in  the  case 
of  organic  acids  and  bases,  —  it  does  apply  with  a  fair  degree  of 
accuracy  to  saturated  solutions  of  electrolytes. 

A  saturated  solution  of  silver  chloride  affords  an  example  of 
such  an  equilibrium.  This  salt  is  practically  completely  ionized 
in  a  saturated  solution,  as  represented  by  the  equation 


Applying  the  law  of  mass  action  to  this  equilibrium,  we  obtain 

CAg*  X  CC1  _  jr 
CAgCl 

Since  the  solution  is  saturated,  the  value  of  CAgci  must  remain 
constant  at  constant  temperature,  and  therefore 

CAg-  X  cci'  =  constant  =  s, 

where  the  product  of  the  ionic  concentrations  s  is  called  the  solu- 
bility or  ionic  product. 

The  equilibria  in  the  above  heterogeneous  system  may  be  repre- 
sented thus  :  — 


(in  solution)      (solid) 

The  solubility  product  for  silver  chloride  at  25°  is  1.56  X  10~10, 
the  ionic  concentrations  being  expressed  in  mols  per  liter.  Hence, 
since  the  two  ions  are  present  in  equivalent  amounts,  a  saturated 
solution  of  silver  chloride  at  25°  must  contain  Vl.56  X  10~1( 
=  1.25  X  10~5  mols  per  liter  of  Ag*  and  Cl'  ions.  In  general,  if 

71  A  < 


434  THEORETICAL  CHEMISTRY 

represents  the  equilibrium  between  an  electrolyte  and  its  products 
of  dissociation  in  saturated  solution,  we  have 

rii          n^ 

Cm     /»  — —       O 

A    '  L    .     —  o. 
A!         A2 

The  solubility  product  may  be  defined  as  the  maximum  product  of 
the  ionic  concentrations  of  an  electrolyte  which  can  exist  at  any  one 
temperature. 

Just  as  the  dissociation  of  a  gaseous  substance  or  of  an  organic 
acid  is  depressed  by  the  addition  of  one  of  the  products  of  dis- 
sociation, so  when  a  substance  with  a  common  ion  is  added  to  the 
saturated  solution  of  an  electrolyte,  the  dissociation  is  depressed 
and  the  undissociated  substance  is  precipitated. 

The  following  example  will  serve  to  illustrate  how  the  solu- 
bility product  of  a  substance  can  be  determined,  and  how  the 
change  in  solubility  due  to  the  addition  of  a  substance  containing 
a  common  ion  may  be  calculated.  The  solubility  of  silver  bromate 
at  25°  is  0.0081  mol  per  liter.  If  we  assume  complete  ionization, 
the  concentration  of  the  ions,  Ag"  and  BrCY  will  be  the  same 
and  equal  to  0.0081  mol  per  liter,  or 

(0.0081)  (0.0081)  =  s. 

The  solubility  in  a  solution  of  silver  nitrate  containing  0.1  mol 
of  Ag'  ions  can  be  calculated  from  the  equation 

(0.0081)2  =  (0.0081  +  0.1  -  x)  (0.0081  -  x), 

where  x  represents  the  amount  of  silver  bromate  thrown  out  of  so- 
lution by  the  addition  of  0.1  mol  of  Ag*  ion.  Since  (0.0081  —  x) 
represents  the  concentration  of  Ag*  and  BrO3'  ions  after  the 
addition  of  the  silver  nitrate,  it  also  represents  the  solubility  of 
silver  bromate  under  similar  conditions.  The  effect  of  adding  a 
solution  of  a  soluble  bromate  containing  0.1  mol  of  Br(V  ion  will 
be  the  same  as  that  produced  by  0.1  mol  of  Ag*  ion. 

The  Basicity  of  Organic  Acids.  The  Ostwald  dilution  law 
holds  strictly  for  all  monobasic  organic  acids,  and  also  for  poly- 
basic  organic  acids  which  are  less  than  50  per  cent  ionized.  The 
neutral  salts  of  these  acids,  however,  are  much  more  highly  ionized, 
and  the  difference  in  conductance  between  two  dilutions  of  a  neu- 
tral salt  of  a  polybasic  acid  is  greater  than  the  difference  in 


ELECTROLYTIC  EQUILIBRIUM   AND   HYDROLYSIS       435 


conductance  between  the  same  dilutions  of  a  neutral  salt  of 
a  monobasic  acid.  Ostwald  *  has  shown  that  it  is  possible 
to  estimate  the  basicity  of  an  organic  acid  from  the  difference 
in  the  equivalent  conductance  of  its  sodium  salt  at  two  different 
dilutions. 

As  the  result  of  a  long  series  of  experiments,  he  found  that  the 
difference  between  the  equivalent  conductance  of  the  sodium  salt 
of  a  monobasic  organic  acid  at  v  =  32  liters  and  at  v  =  1024  liters 
is  approximately  10  units.  Similarly,  the  difference  for  a  dibasic 
acid  between  the  same  dilutions  is  20  units,  and  for  an  n-basic 
acid  the  difference  is  10  n.  Hence,  to  estimate  the  basicity  of  an 
organic  acid,  the  equivalent  conductance  of  its  sodium  salt  at  v  = 
32  liters  and  at  v  =  1024  liters  is  determined;  then,  if  A  is  the  dif- 
ference between  the  values  of  the  conductance  at  the  two  dilutions, 

the  basicity  will  be  n  =  -^.  • 

The  following  table  gives  the  values  of  A  and  n  for  the  sodium 
salts  of  several  typical  organic  acids :  — 


Acid. 

A 

n 

Formic 

10  3 

1 

Acetic 

9  5 

1 

Propionic 

10  2 

1 

Benzoic 

8.3 

1 

Quininic                                                                      .    . 

19.8 

2 

Pyridine-tricarboxylic  (1,  2,  3)                          

31.0 

3 

Pyridine-tricarboxylic  (1,  2,  4)  f  
Pyridine-tetracarboxylic  ...              

29.4 
41.8 

3 
4 

Pyridine-pentacarboxylic  

50.1 

5 

Influence  of  Substitution  on  lonization.  Attention  has  already 
been  called  to  the  marked  difference  in  the  strength  of  acetic 
acid  produced  by  the  replacement  of  the  hydrogen  atoms  of  the 
methyl  group  by  chlorine.  In  the  accompanying  table  the  ioniza- 
tion  constants  for  various  substitution  products  of  acetic  acid 
are  given :  — 

*  Zeit.  phys.  Chem.,  i,  105  (1887);  2,  902  (1888). 


436 


THEORETICAL  CHEMISTRY 


Acid. 


Acetic,  CHsCOOH 

Propionic,  CH3CH2COOH. . . . 
Chloracetic,  CH2C1COOH. . . . 
Bromacetic,  CH2BrCOOH. . . . 
Cyanacetic,  CH2CNCOOH. . . 

Glycollic,  CH2OHCOOH 

Phenylacetic,  C6H5CH2COOH 
Amidoacetic,  CH2NH2COOH. 


lonization 
Constant  (25°). 


0.000018 

0.000013 

0.00155 

0.00138 

0.00370 

0.000152 

0.000056 

3.4X  10-1C 


This  table  affords  an  interesting  illustration  of  the  influence  of 
different  substituents  on  the  strength  of  acetic  acid.  Thus,  the 
activity  of  the  acid  is  increased  by  the  replacement  of  alkyl  hydro- 
gen atoms  by  Cl,  Br,  CN,  OH,  or  C6H5,  while  the  substitution  of 
the  CH3  or  NH2  groups  diminishes  its  activity.  If  we  assume 
that  the  substituents  retain  their  ion-forming  capacity  on  enter- 
ing into  the  molecule  of  acetic  acid,  these  differences  in  activity 
can  be  readily  explained.  Thus,  Cl,  Br,  CN,  and  OH  tend  to 
form  negative  ions,  and  hence  increase  the  negative  character  of 
the  group  into  which  they  enter.  On  the  other  hand,  basic  groups, 
such  as  NH2,  diminish  the  tendency  of  the  group  into  which  they 
.enter  to  yield  negative  ions. 

The  influence  of  an  alkyl  residue  on  the  strength  of  an  organic 
acid  is  conditioned  by  its  distance  from  the  carboxyl  group.  This 
is  well  illustrated  by  the  ionization  constants  of  propionic  acid 
and  some  of  its  derivatives. 


Acid. 

Ionization 
Constant  (25°). 

Propionic  acid,  CH3CH2COOH  .  . 

0  0000134 

Lactic  acid,  CH3CHOHCOOH 

0  000138 

/8-oxypropionic  acid,  CH2OHCH2COOH      . 

0  0000311 

The  effect  of  the  OH  group  in  the  a-position  is  seen  to  be  much 
more  marked  than  when  it  occupies  the  /3-position. 

The  position  of  a  substituent  in  the  benzene  nucleus  exerts  a 
marked  influence  on  the  strength  of  the  derivatives  of  benzoic 


ELECTROLYTIC  EQUILIBRIUM  AND  HYDROLYSIS      437 

acid.     The  ionization  constants  of  benzoic  acid  and  the  three 
chlorbenzoic  acids  are  given  in  the  following  table:  — 


Acid. 

Ionization 
Constant  (25°). 

Benzoic  acid,  C6H5COOH  

0  000073 

o-Chlorbenzoic  acid,  C6H4C1COOH  

0  00132 

m-Chlorbenzoic  acid   CeEUClCOOH 

0  000155 

p-Chlorbenzoic  acid,  C6H4C1COOH   . 

0  000093 

When  the  halogen  enters  the  ortho-position,  the  strength  of  the 
acid  is  greatly  augmented,  while  in  the  meta-  and  para-  positions 
the  effect  is  much  smaller,  meta-chlorbenzoic  acid  being  stronger 
than  para-chlorbenzoic  acid.  It  is  a  general  rule  that  the  influence 
of  substituents  is  always  greatest  in  the  ortho-position,  and  least 
in  the  meta-  and  para-  positions,  the  order  in  the  two  latter  being 
uncertain. 

Hydrolysis.  When  a  salt  formed  by  a  weak  acid  and  a  strong 
base,  such  as  sodium  carbonate,  is  dissolved  in  water,  the  solution 
shows  an  alkaline  reaction,  while  on  the  other  hand,  when  a  salt 
formed  by  a  strong  acid  and  a  weak  base,  such  as  ferric  chloride, 
is  dissolved  in  water,  the  solution  shows  an  acid  reaction. 

The  process  which  takes  place  in  the  aqueous  solution  of  a  salt, 
causing  it  to  react  alkaline  or  acid,  is  termed  hydrolysis  or  hydro- 
lytic  dissociation.  If  MA  represents  a  salt,  in  which  M  is  the  basic 
and  A  is  the  acidic  portion,  then  the  hydrolytic  equilibrium  may 
be  represented  by  the  equation 

MA  +  H20  <=±  MOH  +  HA. 

If  the  base  formed  is  insoluble  or  undissociated  and  the  acid  is 
dissociated,  the  solution  will  react  acid.  If  the  acid  formed  is 
insoluble  or  undissociated  and  the  base  is  dissociated,  the  solution 
will  react  alkaline.  Finally,  if  both  base  and  acid  are  insoluble 
or  undissociated,  the  salt  will  be  completely  transformed  into  base 
and  acid,  and,  as  there  will  be  no  excess  of  either  H'  or  OH7  ions, 
the  solution  will  remain  neutral. 

It  is  evident,  then,  that  hydrolysis  is  due  to  the  remove!  of 
either  one  or  both  of  the  ions  of  water  by  the  ions  of  the  salt  to 


438  THEORETICAL  CHEMISTRY 

form  undissociated  or  insoluble  substances.  As  fast  as  the  ions 
of  water  are  removed,  the  loss  is  made  good  by  the  dissociation  of 
more  water,  until  eventually  a  condition  of  equilibrium  is  estab- 
lished. The  conditions  governing  hydrolytic  equilibrium  may  be 
determined  from  a  knowledge  of  the  solubility  or  ionic  constant 
of  the  substances  involved.  Thus,  if  the  product  of  the  concen- 
trations of  the  ions  M"  and  OH'  exceeds  that  which  can  exist  in 
pure  water,  then  some  undissociated  or  insoluble  substance  will 
be  formed.  This  will  disturb  the  equilibrium  of  H*  and  OH' 
ions,  and  a  further  dissociation  of  water  must  occur  until  the 
ionic  product  of  water  is  just  reached. 

If  now  the  ions  H*  and  A'  do  not  unite  to  form  undissociated 
acid,  the  presence  of  an  excess  of  H"  ions  will  disturb  the  equi- 
librium between  pure  water  and  its  products  of  dissociation;  or, 
since 

CH*  X  COH'  =  Sn2o, 
the  concentration  of  OH'  ions  present,  when  CH-  represents  the 

total  concentration  of  H*  ions,  will  be  -3fi . 

CH» 

A  similar  readjustment  will  take  place  when  an  undissociated 
or  insoluble  acid  and  a  dissociated  base  are  formed. 

We  may  now  proceed  to  consider  three  different  cases  of  hydroly- 
sis, viz.,  when  the  reaction  is  caused  (1)  by  the  base,  (2)  by  the 
acid,  and  (3)  by  both  base  and  acid. 

CASE  I.  The  formation  of  an  undissociated  or  insoluble  base  is  pri- 
marily the  cause  of  the  hydrolysis,  the  acid  formed  being  dissociated. 

Let  the  hydrolytic  equilibrium  be  represented  by  the  equation 

MA  +  H2O  <=*  MOH  +  HA. 

The  reaction  will  proceed  in  the  direction  of  the  upper  arrow  until 
the  product,  CM-  X  COH',  exceeds  that  which  can  exist  in  the  ab- 
sence of  an  undissociated  base.  When  equilibrium  is  established, 
we  have 

final  CM-  X  final  COH-  =  #MOH  X  CMOH  formed,  (1) 

or  if  the  base  formed  is  practically  insoluble,  the  equilibrium  equa- 
tion simplifies  to  the  form 

final  CM'  X  final  COH'  =  SMOH,  (2) 


ELECTROLYTIC  EQUILIBRIUM  AND  HYDROLYSIS      439 

where  SMOH  is  the  solubility  product  of  the  base.  The  condition 
of  equilibrium  represented  by  the  equation 

CH-  X  COH'  =  SHZO 

must  be  fulfilled.  It  follows  that  the  final  concentration  of  the 
OH'  ions  will  be  the  quotient  obtained  by  dividing  the  ionic  product 
for  water,  at  the  temperature  of  the  experiment,  by  the  final  con- 
centration of  the  H*  ion,  this  latter  being  wholly  dependent  upon 
the  extent  of  the  reaction  and  the  degree  of  ionization  of  the  acid 
formed.  If  the  degree  of  hydrolysis  of  the  salt  be  represented  by 
x,  and  the  degree  of  dissociation  of  the  unhydrolyzed  portion  of 
the  salt  be  denoted  by  as)  then,  if  one  mol  of  salt  be  dissolved 
in  V  liters  of  solution,  the  final  concentration  of  M"  ions  will  be 

- — y and  the  final  concentration  of  the  undssiociated  base 

will  be  y>  The  total  acid  formed  will  be  y,  and  if  aa  denotes 
the  degree  of  dissociation  of  the  acid,  the  concentration  of  the  H* 

1* 

ions  will  be  aay  •  Substituting  these  values  in  equations  (1)  and 
(2),  we  obtain 

ds  (1  -  X)      SR20  ~  X  (   . 

~V~        ~  =  A-MOHXy' 

day 

and 

(4) 


Simplifying  equations  (3)  and  (4),  we  have 
&         da  _     SHZO      . 

aM\  T7  *  -  ~   IF  -   ~~ 
-  x)  V    as      AMOH 

and 


—  —  .      . 

(1  —  x)    as       SMOH 

From  equations  (5)  and  (6)  it  appears  that  the  constant  of  hydrol- 
ysis can  be  found  either  from  the  ionic  product  for  water  and  the 


440  THEORETICAL  CHEMISTRY 

ionization  constant  of  the  base,  or  from  the  ionic  product  for 
water  and  the  solubility  product  of  the  base.  Furthermore,  if  the 
base  formed  is  insoluble,  equation  (6)  shows  that  the  degree  of 
hydrolysis,  x,  is  independent  of  the  dilution  of  the  salt,  V. 

CASE  II.  The  formation  of  an  undissociated  or  insoluble  add 
is  primarily  the  cause  of  the  hydrolysis,  the  base  formed  being  dis-^ 
sociated.  In  this  case  hydrolysis  takes  place  until  the  product 
CH*  X  CA'  exceeds  that  which  can  exist  in  the  absence  of  undis- 
sociated acid.  When  equilibrium  is  established,  we  have 

final  CH-  X  final  CA'  =  KHA  X  CHA  formed,  (7) 

or  if  the  acid  formed  is  practically  insoluble,  the  equilibrium  equa- 
tion simplifies  to  the  form 

final  CH-  X  final  CA'  =  SHA.  (8) 

Since  the  final  CH'  =  SHZO  -*-  final  COH',  we  have,  final  CA'  =    "      —  -, 


yj» 

final  COH'  =  ctb  y  ,  where  ah  is  the  degree  of  dissociation  of  the  base 

formed,  and  the  final  CHA  = 
tions  (7)  and  (8),  we  obtain 


formed,  and  the  final  CHA  =      •    Substituting  these  values  in  equa- 


(9) 
•T 

and 


Oity 


Simplifying  equations  (9)  and  (10),  we  have 


and 


SH.O  _    jf 
~ 


(1  —  x)    as 


It  is  evident  from  equations  (11)  and  (12),  that  the  constant  of 
hydrolysis  can  be  found  either  from  the  ionic  product  for  water  and 


ELECTROLYTIC  EQUILIBRIUM  AND  HYDROLYSIS      441 

the  ionization  constant  of  the  acid,  or  from  the  ionic  product  for 
water  and  the  solubility  product  of  the  acid. 

CASE  III.  The  formation  of  an  acid  and  a  base,  both  being 
slightly  dissociated,  is  the  cause  of  the  hydrolysis. 

In  this  case  let  us  assume  that  KHA  is  smaller  than 


Since  the  final  COH'  =  XMQH  x  CMOH   and  since  both  HA  and  MOH 

CM* 

are   slightly  dissociated,  we  may  write  CHA  =  CMOH  =  y,  and 

as  (1  -  x) 
CA'  =  CM-  =  --  y  — 

Substituting  these  values  in  equation  (7),  we  obtain 

x 


Simplifying  equation  (13),  we  obtain 


(1-s)    c?      KHAX^MOH 

From  equation  (14)  we  see  that  the  constant  of  hydrolysis  can  be 
found  from  the  ionic  product  for  water  and  the  ionization  constants 
of  the  acid  and  the  base.  If  both  acid  and  base  are  practically 
insoluble,  the  reaction  will  be  complete  at  all  dilutions. 

As  an  illustration  of  the  application  of  the  foregoing  equations, 
we  may  take  the  calculation  of  the  degree  of  hydrolytic  dissociation 
of  potassium  cyanide  in  0,1  molar  solution  at  25°.  Potassium 
cyanide  being  a  salt  of  a  weak  acid,  the  degree  of  hydrolysis  can 
be  calculated  by  means  of  the  equation 

X2  Cib  _   SH20    _      K 

'       ~ 


Since  at  25°,  KHA  =  13  X  10"10    and  sH2o  =  (0.91  X  10~7),   we 
have 

SH,O       (0.91  XlO-7) 
13  X  10-10   ' 


442  THEORETICAL  CHEMISTRY 

and  since  in  dilute  solution  aa  =  <*&  =  1,  we  have 

x2  (0.91  X  1Q-7)2 

(1  -  x)  - 10         13  X  ID"10 
or 

x  =  0.00798. 

Experimental  Determination  of  Hydrolysis.  The  degree  of 
hydrolysis  can  be  determined  experimentally  in  several  different 
ways.  A  very  convenient  method  is  that  based  upon  measure- 
ments of  electrical  conductance.  When  a  salt  reacts  hydrolyti- 
cally  with  one  mol  of  water,  the  limiting  value  of  its  equivalent 
conductance  will  be  A  A  +  A^,  where  A  A  and  AS  denote  the  equiv- 
alent conductances  of  the  acid  and  base  formed.  If  A  is  the  equiv- 
alent conductance  of  the  unhydrolyzed  salt,  and  A/t  is  the  actual 
conductance  of  the  salt  at  the  same  dilution,  then  the  increase  in 
conductance  corresponding  to  a  degree  of  hydrolysis  x  will  be 
Ah  —  A.  The  value  of  A  may  be  found  by  determining  the 
conductance  of  the  salt  in  the  presence  of  an  excess  of  one  of  the 
products  of  hydrolysis  and  deducting  from  it  the  conductance  of 
the  substance  added.  Since  if  the  hydrolysis  were  complete,  the 
equivalent  conductance  would  be  AA  +  AB  —  A,  we  have 

AA-  A 
"  AA+  AB-  A' 

all  conductances  being  measured  at  the  same  dilution  and  the 
same  temperature.  The  following  example  will  illustrate  the 
use  of  this  equation :  —  At  25°,  the  equivalent  conductance  of  an 
aqueous  solution  of  aniline  hydrochloride  is  118.6,  the  dilution 
being  99.2  liters.  The  equivalent  conductance  in  the  presence  of 
an  excess  of  aniline  is  103.6,  while  the  equivalent  conductance  of 
hydrochloric  acid  at  the  same  dilution  is  411.  The  conductance 
of  pure  aniline  is  so  small  as  to  be  negligible.  Substituting  these 
values  in  the  equation,  we  find 

118.6  -  103.6      nnA. 
'-  411  -  103.6    =  °'°488- 

Lunden  *  has  shown  how  this  method  may  be  extended  to  cases 
where  both  acid  and  base  are  slightly  dissociated. 

*  Jour.  chim.  phys.,  5, 145,  574  (1907). 


ELECTROLYTIC  EQUILIBRIUM  AND  HYDROLYSIS      443 

The  lonization  Constant  of  Water.  One  of  the  most  accurate 
methods  known  for  the  determination  of  the  ionization  constant 
of  water  is  based  upon  measurements  of  the  degree  of  hydrolytic 
dissociation  of  different  salts.  Thus  Shields  *  found  that  a  0.1 
molar  solution  of  sodium  acetate  is  0.008  per  cent  hydrolyzed  at 
25°.  We  may  consider  the  salt,  as  well  as  the  sodium  hydroxide 
formed  from  its  hydrolysis,  to  be  completely  dissociated  at  this 
dilution.  The  ionization  constant  of  the  acetic  acid  formed  is 
0.000018  at  25°.  Solving  equation  (11)  (on  page  440)  for  SH.O, 
and  remembering  that  as  =  ab  =  1,  we  have 


Substituting  the  above  values  in  this  expression,  we  obtain 


SHlo  =  0.000018  -      _  10  =  U6  X  10-, 


and  since  the  ions,  H*  and  OH',  are  present  in  equivalent  amounts, 
we  have 


CH-  =  COH'  =  Vl.16  X  10-14  =  1.1  X  10~7  mol  per  liter. 

Kohlrausch  obtained  from  his  measurements  of  the  conductance 
of  pure  water  at  25°,  CH-  =  COH'  =  1.05  X  10~7  mol  per  liter  (see 
p.  415). 

PROBLEMS. 

1.  At  25°  the  specific  conductance  of  butyric  acid  at  a  dilution  of 
64  liters  is  1.812  X  10"4  reciprocal  ohms.     The  equivalent  conductance 
at  infinite  dilution  is  380  reciprocal  ohms.     What  is  the  degree  of  ioniza- 
tion and  the  concentration  of  H*  ions  in  the  solution?     What  is  the  ioni- 
zation constant  of  the  acid? 

Ans.  a  =  0.0305,  CH-  =  4.765  X  10~4mol  per  liter,  K  =  1.5  X  1Q-5. 

2.  The  heat  of  neutralization  of  nitric  acid  by  sodium  hydroxide  is 
13,680  calories,  and  of  dichloracetic  acid,   14,830  calories.     When  one 
equivalent  of  sodium  hydroxide  is  added  to  a  dilute  solution  containing 
one  equivalent  of  nitric  acid  and  one  equivalent  of  dichloracetic  acid, 
13,960  calories  are  liberated.    What  is  the  ratio  of  the  strengths  of  the 
two  acids?  Ans.  HN03  :  CHC12COOH  ::  3.1  :  1. 

*  Zeit.  phys.  Chem.,  12,  167  (1893). 


444  THEORETICAL  CHEMISTRY 

3.  For  potassium  acetate  we  have  the  following  data:  — 


r 

A«(18°) 

2 

67.1 

10 

78.4 

100 

87.9 

1000 

91.9 

and  ZOQ  =  64.67,  and  1M  =  35.    Compare  the  constants  obtained  by  the 

K'  CH3COO' 

Ostwald,  Rudolphi,  and  van't  Hoff  dilution  laws. 

4.  The  ionization  constant  of  a  0.05  molar  solution  of  acetic  acid  is 
0.0000175  at  18°,  and  0.00001624  at  52°.    Calculate  the  heat  of  ionization 
of  the  acid.    To  what  temperature  does  this  value  correspond  ? 

Ans.  416  calories  at  35°. 

5.  At  20°  the  specific  conductance  of  a  saturated  solution  of  silver 
bromide  was  1.576  X  10~6  reciprocal  ohms,  and  that  of  the  water  used 
was   1.519  X  10~6  reciprocal  ohms.     Assuming  that  silver  bromide  is 
completely  ionized,  calculate  the  solubility  and  the  solubility  product  of 
silver  bromide,  having  given  that  the  equivalent  conductances  of  potas- 
sium bromide,  potassium  nitrate,  and  silver  nitrate  at  infinite  dilution 
are  137.4,  131.3,  and  121  reciprocal  ohms  respectively. 

Ans.  CAgBr  =  4.49  X  10~7  mol  per  liter,  SAgBr  =  2.02  X  10~13. 

6.  The  solubility  of  silver  cyanate  at  100°  is  0.008  mol  per  liter.     Cal- 
culate the  solubility  in  solution  of  potassium  cyanate  containing  0.1  mol 
of  K'  ions.  Ans.  6.4  X  10~4  mol  per  liter. 

7.  Calculate  the  degree  of  hydrolytic  dissociation  of  a  0.1  molar  solu- 
tion of  ammonium  chloride,  having  given  the  following  data:  —  aa  =  0.86, 
«a  =  0.87,  tf  NH4OH  =  0.000023,  and  SH2o  =  (0.91  X  10~7)2  at  25°. 

Ans.  x  =  0.006  per  cent. 

8.  In  the  reaction  represented  by  the  equation 

MA3  +  3  H20  =  M  (OH),  +  3  HA, 

the  base  formed  is  insoluble.     Derive  an  expression  for  the  constant  of 
hydrolysis. 

V  S3H2O  (3z)3         aa3 

A  VI  O          fi  —  —          __  _  i  ^ 

-n/t'O.    -ivp  —  —   ,  \   o 

SM(OH)3        (1  -  X)VZ      a, 

9.  The  equivalent  conductance  of  aniline  hydrochloride  at  a  dilution 
of  197.6  liters  is  126.7  reciprocal  ohms,  at  25°.    The  equivalent  con- 


ELECTROLYTIC  EQUILIBRIUM  AND  HYDROLYSIS       445 

ductance  of  aniline  hydrochloride  in  the  presence  of  an  excess  of  aniline 
is  106.6;  and  the  equivalent  conductance  of  hydrochloric  acid  at  the 
same  dilution  is  415.  If  the  conductance  of  pure  aniline  is  negligible, 
calculate  the  degree  of  hydrolytic  dissociation  and  the  constant  of  hydrol- 
ysis, assuming  as  =  aa  =  1. 

Am.  x  =  6.52  per  cent,  Kh  =  2.30  X  10~5. 

10.  The  hydrolysis  constant  of  aniline  is  2.25  X  10~5,  and  the  ioniza- 
tion  constant  is  5.3  X  10~10.  Calculate  the  concentration  of  the  H" 
and  OH'  ions  in  water,  Ans.  CCH-  =  CCH'  =  1.09  X  10~7. 


CHAPTER  XX. 
ELECTROMOTIVE  FORCE. 

Galvanic  Cells.  Since  the  year  1800,  when  Volta  invented 
his  electric  pile,  many  different  forms  of  galvanic  cell  have  been 
introduced. 

It  is  not  our  purpose  to  give  a  detailed  account  of  these  cells, 
but  rather  to  give  a  brief  outline  of  the  theories  which  have  been 
advanced  in  explanation  of  the  electromotive  force  developed  in 
such  cells.  When  two  metallic  electrodes  are  immersed  in  a  solu- 
tion of  an  electrolyte,  a  current  will  flow  through  a  wire  connect- 
ing the  electrodes,  provided  the  two  metals  are  dissimilar,  or  that 
a  difference  exists  between  the  solutions  surrounding  the  electrodes. 
An  electric  current  can  be  obtained  from  a  combination  of  two 
different  metals  in  the  same  electrolyte,  from  two  different  metals 
in  two  different  electrolytes,  from  the  same  metal  in  different  elec- 
trolytes, or  from  the  same  metal  in  two  different  concentrations  of 
the  same  electrolyte. 

In  order  that  the  electromotive  force  of  the  combination  shall 
remain  constant,  it  is  necessary  that  the  chemical  changes  involved 
in  the  production  of  the  current  shall  neither  destroy  the  difference 
between  the  electrodes,  nor  deposit  upon  either  of  them  a  non- 
conducting substance.  A  galvanic  combination  which  fulfils 
these  conditions  very  satisfactorily  is  the  Daniell  cell.  This  cell 
consists  of  zinc  and  copper  electrodes  immersed  in  solutions  of 
their  salts,  as  represented  by  the  scheme 

Zn  -  Sol.  of  ZnS04||  Sol.  of  CuS04  -  Cu, 

in  which  the  two  vertical  lines  indicate  a  porous  partition  separat- 
ing the  two  solutions.  When  the  zinc  and  copper  electrodes  are 
connected  by  a  wire,  a  current  of  positive  electricity  passes  from 
the  copper  to  the  zinc  along  the  wire.  Zinc  dissolves  from  the  zinc 
electrode,  an  equivalent  amount  of  copper  being  displaced  from 

446 


ELECTROMOTIVE  FORCE  "447 

the  solution  and  deposited  simultaneously  on  the  copper  electrode. 
As  long  as  only  a  moderate  current  flows  through  the  cell,  the 
original  nature  of  the  electrodes  is  not  modified,  the  only  change 
which  occurs  being  the  gradual  dilution  of  the  copper  sulphate, 
owing  to  the  separation  of  copper  and  its  replacement  by  zinc. 
If  the  loss  of  copper  sulphate  is  replaced,  the  electromotive  force 
of  the  cell  will  remain  constant.  If,  after  the  cell  is  assembled  no 
current  be  allowed  to  flow,  the  copper  sulphate  will  slowly  diffuse 
into  the  solution  of  zinc  sulphate,  and  metallic  copper  will  ulti- 
mately be  deposited  on  the  zinc  electrode.  In  this  way  miniature, 
local  galvanic  cells  will  be  formed  on  the  surface  of  the  zinc,  and 
the  metal  dissolves  as  though  the  main  circuit  were  closed.  Until 
this  deposition  takes  place,  the  cell  may  be  left  on  open  circuit 
without  danger  of  deterioration.  Unless  chemically  pure  zinc  is 
used,  local  action  is  likely  to  occur,  owing  to  the  formation  of  local 
galvanic  couples  between  the  impurities  in  the  electrode,  —  chiefly 
iron,  —  and  the  zinc.  This  action  may  be  prevented  by  amalga- 
mating the  zinc  electrode.  In  this  process  the  mercury  dissolves 
the  zinc  and  not  the  iron,  a  uniform  surface  of  the  former  metal 
being  produced. 

An  interesting  experiment  due  to  Ostwald  *  illustrates  the  con- 
ditions essential  to  the  continuous  production  of  an  electric  current. 
Two  electrodes,  one  of  amalgamated  zinc  and  the  other  of  platinum, 
are  each  immersed  in  a  solution  of  potassium  sulphate,  the  two 
solutions  being  separated  by  a  porous  cup.  When  the  two  elec- 
trodes are  connected  by  means  of  a  wire,  no  permanent  current 
passes.  An  inappreciable  quantity  of  zinc  goes  into  solution, 
since  any  current  must  necessarily  first  liberate  potassium  at  the 
platinum  electrode,  the  potassium  thus  set  free  reacting  with  the 
water.  This  process  requires  the  expenditure  of  more  energy 
than  the  solution  of  the  zinc  supplies.  If  sulphuric  acid  is  added 
to  the  compartment  containing  the  zinc,  the  condition  of  the 
system  will  be  unchanged,  the  zinc  remaining  undissolved.  If, 
on  the  other  hand,  a  few  drops  of  sulphuric  acid  are  added  to  the 
compartment  containing  the  platinum  electrode,  bubbles  of 
hydrogen  will  appear  and  the  zinc  will  dissolve  with  the  simulta- 
*  Phil.  Mag.  [5],  32,  145  (1891). 


448  THEORETICAL  CHEMISTRY 

neous  development  of  an  electric  current.  This  experiment  shows 
that  in  order  that  positively  charged  ions  may  enter  a  solution, 
an  equivalent  amount  of  negatively  charged  ions  must  be  intro- 
duced, or  an  equivalent  amount  of  positively  charged  ions  must 
be  removed. 

Reversible  Cells.  Galvanic  cells  are  either  reversible  or  non- 
reversible,  according  as  the  processes  taking  place  within  them  can 
be  reversed  or  not.  If  we  disregard  the  slow  processes  of  diffusion, 
the  Daniell  cell  may  be  taken  as  an  example  of  an  almost  perfect 
reversible  element.  If  an  electromotive  force  slightly  less  than 
that  of  the  cell  be  applied  to  it  in  the  reverse  direction,  the  current 
within  the  cell  will  flow  from  the  zinc  to  the  copper  electrode  as 
usual.  On  the  other  hand,  if  the  external  electromotive  force 
slightly  exceeds  that  of  the  cell,  the  current  within  the  cell  will 
flow  in  the  reverse  direction,  zinc  being  deposited  and  copper 
dissolved. 

Any  cell  from  which  gas  is  evolved  is  non-reversible,  since  the 
passage  of  a  current  in  the  reverse  direction  cannot  restore  the 
cell  to  its  original  condition. 

Relation  between  Chemical  Energy  and  Electrical  Energy. 
Helmholtz  and  Thomson  were  the  first  to  propose  a  theory  of  the 
action  of  the  reversible  cell.  According  to  this  theory  the  energy 
of  the  chemical  process  taking  place  within  the  cell  was  considered 
to  be  completely  transformed  into  electrical  energy.  It  was  soon 
shown  that  this  theory  is  inadequate,  since,  with  the  exception  of 
the  Daniell  cell,  the  chemical  energy  is  not  equivalent  to  the  elec- 
trical energy  produced.  Subsequently,  Gibbs*  and  Helmholtz  f 
showed  independently  that  only  in  those  cells  in  which  the  elec- 
tromotive force  does  not  vary  with  the  temperature,  is  the  chem- 
ical energy  completely  transformed  into  electrical  energy.  They 
also  derived  an  equation  expressing  the  relation  between  the 
chemical  and  electrical  energies  in  any  reversible  cell.  Let  us 
imagine  a  reversible  element  in  which  an  amount  of  heat  q,  is  either 
liberated  or  absorbed,  when  one  faraday  of  electricity  has  passed 
through  the  cell.  Let  the  cell  be  immersed  in  a  bath,  which  is  so 

*  Proc.  Conn.  Acad.,  3,  501  (1878). 

t  Sitzungsbericht.,  Ber.  Akad.,  22,  825  (1882). 


ELECTROMOTIVE  FORCE  449 

arranged  that  the  temperature  of  the  cell  can  be  maintained  con- 
stant under  any  working  conditions.  If  the  chemical  process 
within  the  cell  is  accompanied  by  an  evolution  or  an  absorption 
of  heat,  then  of  necessity,  heat  energy  must  be  removed  or 
supplied  in  order  to  maintain  the  temperature  of  the  system 
constant.  It  is  evident  that  this  will  involve  a  corresponding 
decrease  or  increase  in  the  electrical  energy  produced  by  thet  cell. 
The  effect  of  the  evolution  or  absorption  of  heat  upon  the 
electrical  energy  of  the  cell  may  be  derived  in  the  following 
manner:  Let  the  cell  be  heated  from  its  initial  temperature  T 
to  the  temperature  (T  +  dT),  and  let  the  corresponding  change 
in  the  electromotive  force  of  the  cell  be  dE.  If  now  the  circuit  be 
closed  and  one  faraday  of  electricity  be  allowed  to  pass  through 
the  cell,  F  (E  +  dE)  units  of  electrical  work  will  be  done.  In 
order  that  the  temperature  of  the  cell  may  not  change,  (q  +  dq) 
units  of  heat  must  be  absorbed.  The  cell  is  now  cooled  to  the 
temperature  T,  at  which  the  electromotive  force  of  the  cell  is  E, 
and  F  units  of  electricity  are  sent  through  the  cell  in  the  reverse 
direction,  thus  increasing  the  energy  of  the  cell  by  FE.  In  order 
to  maintain  the  temperature  of  the  cell  unchanged,  q  units  of 
heat  must  be  removed.  If  the  cell  is  completely  reversible,  when 
this  cycle  of  operations,  is  completed,  it  will  be  restored  to  its 
original  condition.  The  total  work  done  during  the  cycle  is 
F  (E  +  dE)  —  FE,  and  the  amount  of  heat  transformed  into  work 
is  (q  +  dq)  —  q-}  therefore,  applying  the  second  law  of  thermo- 
dynamics, we  have 

dq  _  FdE  _  dT 
~q  =       q       =  T  ' 

or 


Since  the  electrical  energy  is  equal  to  FE,  the  relation  between 
this  and  Q,  the  chemical  energy  of  the  cell,  expressed  in  calories, 
becomes 

FE  =  Q  +  q.  (2) 


450  THEORETICAL  CHEMISTRY 

Substituting  in  equation  (2)  the  value  of  q  given  in  equation  (1), 
we  obtain 


or 

Q         dE 


When  -y™  =  0,  E  becomes  equal  to  ^  ,  or,  when  the  temperature 

coefficient  of  the  cell  is  zero,  the  electrical  energy  is  equal  to  the 
chemical  energy.  This  is  true  of  the  Daniell  cell,  which  has  an 
extremely  small  temperature  coefficient. 

For  cells  in  which  the  electromotive  force  varies  appreciably 
with  the  temperature,  it  is  possible  to  calculate  the  value  of  the 
electromotive  force  at  any  temperature  by  means  of  the  Gibbs- 
Helmholtz  equation,  provided  the  temperature  coefficient  is  known. 
In.  the  Grove  gas  cell,  E  =  1.062  and  Q  =  34,200  calories,  hence 

TdE  -  1  062  -          34'200  -0  418 

1  df  96,540  X  0.2394  " 

The  value  determined  by  direct  experiment  is  —  0.416  volt.  The 
Gibbs-Helmholtz  equation  shows  that  the  amount  of  heat  accom- 
panying a  chemical  process  does  not  alone  furnish  a  measure  of 
the  electrical  energy  which  may  be  obtained  from  it,  since  the 
heat  which  is  absorbed  from  the  surrounding  medium  may  also 
be  transformed  into  electrical  energy,  or  the  output  of  electrical 
energy  may  be  less  than  the  heat  evolved  by  the  chemical 
reaction  within  the  cell. 

Solution  Pressure.  It  is  a  familiar  fact  that  water  has  a 
tendency  to  assume  the  form  of  vapor,  and  if  the  vapor  be  contin- 
ually removed  from  its  surface,  a  definite  mass  of  water  will  grad- 
ually be  completely  transformed  into  the  state  of  vapor.  The 
pressure  of  the  vapor  at  any  one  temperature  is  a  measure  of  the 
tendency  of  water  to  undergo  this  transformation.  This  tendency 
of  water  to  assume  another  form  than  that  in  which  it  actually 
exists,  is  typical  of  all  substances.  Attention  has  already  been 
directed  to  this  fact  in  connection  with  the  application  of  the  law 


ELECTROMOTIVE  FORCE  451 

of  mass  action  to  heterogeneous  equilibria.  It  was  then  pointed 
out  that  all  solids  have  a  definite  vapor  pressure  at  a  definite 
temperature,  which  is  independent  of  the  amount  of  solid  present. 
When  a  solid,  such  as  cane  sugar,  is  brought  in  contact  with  water, 
it  tends  to  pass  into  solution.  This  tendency  is  constant  at 
constant  temperature,  since  the  active  mass  of  the  solid  is  constant. 
From  the  close  analogy  between  the  vapor  state  and  the  dissolved 
state,  the  tendency  of  a  solid  to  pass  into  solution  is  termed  the 
solution  pressure.  A  dissolved  solid,  on  the  other  hand,  also  shows 
a  tendency  to  separate  from  the  solution  as  the  concentration 
is  increased.  When  the  solution  becomes  supersaturated,  the 
tendency  of  the  solute  to  separate  in  the  solid  form  is  greater  than 
the  tendency  of  the  solid  to  dissolve.  It  is  evident  from  these 
considerations  that  the  pressure  exerted  by  the  dissolved  solid 
is  its  osmotic  pressure,  and  whether  the  solid  will  dissolve  or 
separate  from  the  solution  depends  upon  whether  the  solution 
pressure  is  greater  or  less  than  the  osmotic  pressure. 

This  conception  of  solution  pressure  was  introduced  by  Nernst,* 
and  in  conjunction  with  the  theory  of  electrolytic  dissociation  it 
has  proved  of  great  value  in  affording  a  much  deeper  insight  into 
the  mechanism  of  the  development  of  differences  in  potential 
within  a  galvanic  cell.  Thus,  when  a  metal  is  dipped  into  water 
it  tends  to  dissolve  owing  to  its  solution  pressure  P  and,  in  con- 
sequence of  this  tendency,  it  sends  a  certain  number  of  positive 
ions  into  solution.  The  solution  thus  becomes  positively  charged, 
and  the  metal,  which  was  initially  neutral,  acquires  a  negative 
charge  due  to  the  loss  of  a  certain  amount  of  positive  electricity. 
This  process  will  cease  when  the  solution  becomes  so  strongly 
charged  with  positive  electricity  that  it  prevents  the  separation 
of  any  more  positive  ions  from  the  metal.  Relatively  few  ions 
leave  the  metal  before  equilibrium  is  established,  since  the  charge 
on  each  ion  is  so  great;  in  fact,  the  concentration  of  metal  ions  in 
the  solution  is  much  too  small  to  be  detected  analytically.  When 
a  metal  is  dipped  into  a  solution  of  one  of  its  salts,  the  conditions 
are  altered.  In  this  case,  the  positive  ions  of  the  metal  already 
present  in  the  solution  oppose  the  entrance  of  more  positive  ions, 
*  Zeit.  phys.  Chem.,  4,  150  (1889). 


452 


THEORETICAL   CHEMISTRY 


and  the  equilibrium  between  these  two  opposing  tendencies  will 
be  conditioned  by  the  relative  values  of  the  solution  pressure  P, 
of  the  metal,  and  the  osmotic  pressure  p,  of  the  ions  of  the  dissolved 
salt. 

It  is  evident  that  the  three  following  conditions  are  possible:  — 

(1)  If  P  >  p,  the  metal  will  continue  to  send  ions  into  the 
solution  until  the  accumulated  charges  in  the  solution  oppose 
further  action.     The  solution  acquires  a  positive  charge  and  the 
metal  a  negative  charge. 

(2)  If  P  <  p,  the  positive  ions  of  the  dissolved  salt  will  sepa- 
rate on  the  metal  until  the  accumulated  charges  oppose  further 
action.     The  metal  acquires  a  positive  charge  and  the  solution  a 
negative  charge. 

(3)  If  P  =  p,  no  action  will  take  place  and  no  difference  of 
potential  will  be  established  between  the  metal  and  the  solution. 
These  three  cases  are  represented  diagrammatically  in  Fig.  95. 


?=P 


-h 


Fig.  96. 

When  equilibrium  is  established  and  the  metal  is  negative  against 
the  solution,  the  metal  is  surrounded  by  a  layer  of  positively 
charged  ions.  This  constitutes  what  is  known  as  a  Helmholtz 
electrical  double  layer.  If  positive  electricity  be  communicated  to 
the  metal,  the  double  layer  will  be  broken  and  more  ions  will 
pass  from  the  metal  into  the  solution,  but  as  soon  as  the  supply 


ELECTROMOTIVE  FORCE 


453 


of  positive  electricity  is  cut  off,  the  double  layer  will  again  be 
formed.  Similarly,  when  the  metal  is  positive  against  the  solu- 
tion, an  electrical  double  layer  will  be  formed,  the  metal  being 
surrounded  by  a  layer  of  negatively  charged  ions. 

The  actual  existence  of  a  Helmholtz  double  layer  has  been 
demonstrated  by  Palmaer.*  In  his  experiments,  Palmaer  allowed 
exceedingly  minute  globules  of  mercury  to  fall  into  a  dilute  solu- 
tion of  mercurous  nitrate  contained  in  a  tall  vessel,  the  bottom 
of  which  was  covered  with  a  layer  of  pure  mercury,  as  shown  in 
Fig.  96.  Since  the  solution  pressure  of  mercury  is  less  than  the 


Fig.  96. 

osmotic  pressure  of  the  Hg*  ions,  each  drop  of  mercury  as  it 
enters  the  solution  will  acquire  a  positive  charge,  and  if  the  theory 
of  the  electrical  double  layer  is  correct,  this  positively  charged 
globule  should  attract  negatively  charged  ions  and  drag  them  down 
through  the  solution.  When  the  globule  reaches  the  mercury  at 
*  Zeit.  phys.  Chem.,  25,  265  (1898);  28,  257  (1899);  36,  664  (1901). 


454  THEORETICAL  CHEMISTRY 

the  bottom  of  the  vessel,  it  will  give  up  its  positive  charge  and  as 
many  Hg"  ions  will  pass  into  solution  as  there  are  NCV  ions  in  the 
double  layer.  The  solution  will  thus  become  more  concentrated 
just  above  the  layer  of  mercury  on  the  bottom  of  the  vessel.  Pal- 
maer's  experiments  showed  that  this  difference  in  concentration 
is  actually  produced,  in  some  cases  the  concentration  in  the  upper 
part  of  the  solution  being  reduced  as  much  as  50  per  cent. 

The  metals  sodium,  potassium,  .  .  .  zinc,  cadmium,  cobalt, 
nickel,  and  iron  are  negative  against  solutions  of  their  salts,  or 
P  >  p.  The  noble  metals  are  generally  positive  against  solutions 
of  their  salts,  or  P  <  p.  The  anions  are,  so  far  as  is  known,  posi- 
tive to  solutions  of  their  salts.  Electrolytic  solution  pressure 
varies  with  the  temperature,  with  the  nature  of  the  solvent,  and 
also  with  the  concentration  of  the  active  substance  in  the  elec- 
trode. 

The  Difference  of  Potential  between  a  Metal  and  a  Solution. 
From  the  foregoing  considerations,  it  is  possible  to  derive  an 
equation  expressing  the  difference  of  potential  between  a  metallic 
electrode  and  a  solution  of  one  of  its  salts. 

Let  us  imagine  one  gram-ion  of  a  metal  to  be  transferred  from 
the  electrolytic  solution  pressure  P,  to  the  osmotic  pressure  p. 
The  osmotic  work  done  will  be 


p 

Integrating  this  expression,  we  have 

p 
Osmotic  work  =  RT  log«  —  • 

The  corresponding  electrical  energy  gained  is  nFir,  where  TT  is  the 
difference  of  potential  between  the  metal  and  the  solution,  F  —  1 
faraday  =  96,540  coulombs,  and  n  is  the  valence  of  the  metal. 
Since  the  osmotic  work  done  is  equivalent  to  the  electrical  energy 
gained,  we  may  equate  these  two  expressions,  as  follows:  — 

nFir  =  RT\oge-> 

or 

RT 


ELECTROMOTIVE  FORCE  455 


96,540  X  n  X  0.4343  X  0.2394  Tlog  p  ' 


Expressing  both  sides  of  equation  (1)  in  electrical  units,  and  trans- 

forming to  Briggsian  logarithms,  we  obtain 

p 
p 

or 

0.0002  „„      P 
*  =  -^-T\og-.  (2) 

For  univalent  ions  at  17°,  we  have 

TT  =  0.0575  log--  (3) 

In  a  galvanic  cell  composed  of  two  metals,  each  immersed  in  a 
solution  of  one  of  its  salts,  a  difference  of  potential  may  be  estab- 
lished (1)  at  the  junction  of  the  two  metals,  (2)  at  the  junction  of 
the  two  solutions,  and  (3)  at  the  points  of  contact  of  the  metals 
with  their  respective  solutions.  If  the  temperature  remains  con- 
stant, (1)  is  negligible,  and  in  general,  (2)  is  exceedingly  small; 
therefore,  the  electromotive  force  of  the  cell  may  be  considered  as 
due  to  the  differences  of  potential  arising  at  the  two  electrodes. 
Assuming  the  temperature  to  be  17°,  the  electromotive  force  of 
the  cell  will  be 

,,  0.0575,      Pi      0.0575,      P2 

E  =  TTi  -  7T2  =  —     -  log  ----  log  —  • 
n         &pi  n         &pz 

The  Measurement  of  Electromotive  Force.  The  value  of  the 
electromotive  force  of  a  cell  may  vary  with  the  conditions  of  meas- 
urement. Since,  according  to  Ohm's  law,  E  =  C  (R  +  r),  where 
R  is  the  resistance  of  the  external  circuit  and  r  is  the  internal 
resistance  of  the  cell,  it  follows  that  the  fall  of  potential  CR,  in 
the  external  circuit,  will  only  be  equal  to  E  when  r  is  negligible 
in  comparison  with  R.  Furthermore,  when  the  circuit  is  closed, 
the  electrodes  of  the  cell  frequently  become  polarized,  owing  to 
the  deposition  of  the  products  of  electrolysis,  and  an  opposing 
electromotive  force  is  set  up. 

To  avoid  these  difficulties,  the  electromotive  force  is  usually 
measured  on  open  circuit  by  the  Poggendorff  compensation  method. 
In  this  method  the  electromotive  force  to  be  measured  is  just 


456 


THEORETICAL  CHEMISTRY 


balanced  by  an  equal  and  opposite  electromotive  force,  so  that  no 
current  passes.  The  arrangement  of  the  apparatus  for  such 
measurements  is  shown  in  Fig.  97.  If  the  two  ends  of  the  wire  AB 


Fig.  97. 

of  a  Wheatstone  bridge  are  connected  to  a  lead  accumulator  C, 
there  will  be  a  uniform  fall  of  potential  along  its  length.  The 
amount  of  fall  along  any  portion  AD  will  be  proportional  to  the 

AD 
length  AD,  and  equal  to  the  fraction  -      of  the  total  fall  of  poten- 


tial along  the  entire  length  of  the  wire.  Now  let  one  terminal  of 
a  cell  whose  electromotive  force  is  less  than  that  of  C  be  con- 
nected to  Aj  and  the  other  terminal  be  connected  through  a 
galvanometer  G,  with  a  sliding  contact  D,  the  two  cells  E  and 
C  working  in  opposition.  A  current  will  flow  through  the  cir- 
cuit AEGD,  and  will  be  indicated  by  the  galvanometer  at  all 
positions,  except  that  at  which  the  fall  of  potential  along  the  wire 
from  A  to  D  is  equal  to  the  electromotive  force  of  E.  Hence  we 
have 

e.m.f.  of  C  :  e.m.f.  of  E  ::  AB  :  AD, 

from  which  the  value  of  the  electromotive  force  of  the  cell  E,  can 
be  calculated.  Since  the  electromotive  force  of  a  lead  accumu- 
lator is  not  quite  constant,  it  is  customary,  after  having  deter- 
mined the  point  D,  to  substitute  a  standard  cell  for  E,  and  balance 


ELECTROMOTIVE  FORCE 


457 


this  against  the  accumulator,  finding  a  new  point  of  balance  D'. 
We  now  have  the  proportion 

e.m.f.  of  C  :  e.m.f.  of  standard  ::  AB  :  AD'. 
Combining  these  two  proportions,  we  obtain 

e.m.f.  of  E  :  e.m.f.  of  standard  : :  AD  :  AD'. 

Instead  of  using  a  galvanometer  as  a  "null  "  instrument  for  indi- 
cating when  the  point  of  balance  has  been  reached,  a  capillary 
electrometer  may  be  employed. 

Standard  Cells.  It  is  apparent  that  the  accuracy  of  all 
measurements  of  electromotive  force  is  dependent  upon  the  cell 
employed  as  a  standard.  Much  time  has  been  devoted  to  the 
study  of  various  reversible  elements  with  a  view  to  establishing  a 
standard  of  electromotive  force.  As  a  result,  we  have  the  com- 
plete specifications  for  two  standard  cells,  either  of  which  may  be 
readily  reproduced. 

(a)  The  Weston,  or  Cadmium  Standard  Cell.  The  most  widely 
used  standard  of  electromotive  force  is  the  so-called  Weston  cell, 
made  up  according  to  the  scheme 

Hg  -  Solution  Hg2SO4 1|  Solution  CdS04  -  Cd. 
A  diagram  of  the  usual  form  of  the  Weston  cell  is  given  in  Fig.  98. 


Fig.  98. 


458  THEORETICAL  CHEMISTRY 

A  short  platinum  wire  is  sealed  through  the  bottom  of  each  limb 
of  the  H-shaped  vessel.  In  one  limb  is  placed  a  small  amount  of 

a  10  to  15  per  cent  cadmium  amalgam,  A;  B  is  a  layer  of  small 
g 

crystals  of  CdSO4«~H20.     In  the  other  limb  is  placed  a  small 
o 

amount  of  pure  mercury,  over  which  is  a  layer,  D,  of  a  paste 
composed  of  solid  mercurous  sulphate  and  a  saturated  solution 
of  cadmium  sulphate.  The  cell  is  then  filled  with  crystals  of 
cadmium  sulphate  and  a  saturated  solution  of  cadmium  sulphate. 
The  two  limbs  of  the  cell  are  closed  with  a  thin  layer  of  paraffin 
E,  cork  F,  and  sealing  wax  G.  If  carefully  prepared,  this  cell  will 
remain  unaltered  for  years  and  will  have  an  electromotive  force 
at  20°  of  1.0183  volts.  In  addition  to  the  fact  that  it  can  be  so 
easily  reproduced,  the  temperature  coefficient  of  the  cell  is  almost 
negligible. 

The  electromotive  force  of  a  Weston  standard  cell  at  any  temper- 
ature t,  is  given  by  the  formula 

e.m.f.  at  t°  =  1.0183  -  0.000038  (t  -  20). 

(b)  The  Clark,  or  Zinc  Standard  Cell.  Until  about  ten  years 
ago,  the  Clark  cell  was  considered  to  be  the  most  trustworthy 
standard  of  electromotive  force.  This  cell  is  made  up  according 
to  the  scheme 

Hg  -  Solution  Hg2S04 1|  Solution  ZnS04  -  Zn. 

The  construction  of  the  cell  is  similar  to  that  of  the  Weston  cell. 
It  may  be  reproduced  with  great  accuracy  and  with  no  more 
trouble  than  the  Weston  cell,  but  its  relatively  large  temperature 
coefficient  renders  it  less  satisfactory.  The  electromotive  force 
of  the  Clark  standard  cell  at  any  temperature  t,  may  be  calculated 
by  means  of  the  formula 

e.m.f.  at  t°  =  1.4328  -  0.00119  (t  -  15)  -  0.00007  (t  -  15)2. 

The  Capillary  Electrometer.  When  pure  mercury  is  covered 
with  sulphuric  acid,  its  surface  tension  is  diminished.  This  may 
be  shown  by  the  following  experiment:  In  a  small  evaporating 
dish  place  about  5  cc.  of  pure  mercury,  and  cover  it  with  a  10  per 


ELECTROMOTIVE  FORCE 


459 


cent  solution  of  sulphuric  acid  to  which  has  been  added  enough 
potassium  dichromate  to  impart  a  light  yellow  color  to  the  so- 
lution. The  globule  of  mercury  will  immediately  flatten  out, 
indicating  that  its  surface  tension  has  diminished.  If  now  the  mer- 
cury be  touched  with  a  piece  of  iron  wire,  it  will  instantly  contract 
until  the  contact  with  the  wire  is  broken;  it  will  then  flatten  out, 
until  it  again  comes  in  contact  with  the  wire,  when  the  globule 
of  mercury  will  once  more  contract.  In  this  way  a  regular  pul- 
sation of  the  mercury  may  be  obtained.  This  interesting  phenom- 
enon was  observed  early  in  the  nineteenth  century  by  Henry,  but 
was  first  satisfactorily  explained  by  Lippmann  *  in  1873.  Lipp- 
mann  showed  that  when  the  globule  of  mercury  is  negatively 
electrified,  its  surface  tension  increases  and  the  drop  shrinks. 
If  sufficient  negative  electricity  is  imparted  to  the  mercury  it 
is  possible  to  restore  the  globule  to  its  original  form.  On  ap- 
plying more  negative  electricity,  the  globule  of  mercury  again 
expands.  When  the  iron  wire  touches  the  globule  it  charges 
it  negatively,  because  when  the  iron  dis- 
solves, it  furnishes  positively  charged  ions 
to  the  solution  and  thus  acquires  a  negative 
charge  which  it  imparts  to  the  mercury.  At 
the  same  tune,  the  chromic  acid  in  the  so- 
lution undergoes  reduction  to  chromium 
sulphate.  Lippmann  concluded  from  his 
experiments  that  the  difference  of  poten- 
tial arises  at  the  surface  of  contact  between 
the  mercury  and  the  solution  of  the  electro- 
lyte, and  that  the  surface  tension  of  the 
mercury  is  a  function  of  the  difference  of 
J  I  potential.  Making  use  of  this  principle  he 
f  A  constructed  the  capillary  electrometer,  a  con- 
VHvA  venient  form  of  which  is  shown  in  Fig.  99. 
The  bulb  A,  through  the  bottom  of  which 
is  sealed  a  platinum  wire,  contains  pure 
mercury  and  dilute  sulphuric  acid  (1  :  6). 
Pure  mercury  is  poured  into  the  other  limb  of  the  electrometer 
*  Pogg.  Ann.,  149,  546  (1873). 


Fig.  99. 


460 


THEORETICAL  CHEMISTRY 


until  it  stands  at  B  in  that  tube,  and  at  C  in  the  capillary  tube. 
Owing  to  the  capillary  depression  of  the  mercury,  C  lies  below  B. 
Electrical  connection  with  the  mercury  at  B  is  established  by 
means  of  a  platinum  wire. 

The  position  of  the  mercury  in  the  capillary  is  determined  by 
its  surface  tension;  if  the  surface  tension  is  increased,  the  mercury 
will  descend;  if  it  is  diminished,  the  mercury  will  ascend.  If  a, 
negative  charge  is  communicated  to  the  mercury  at  B,  the  surface 
tension  will  be  increased  and  the  meniscus  will  descend ;  if  a  posi- 
tive charge  is  imparted  to  the  mercury  at  B,  the  surf  ace  tension 
will  be  diminished  and  the  meniscus  will  ascend. 

The  amplitude  of  the  movement  of  the  meniscus  is  an  inverse 
function  of  the  diameter  of  the  capillary  tube.     If  the  meniscus 
be  observed   through   a   microscope 
provided  with  an  eye-piece  microm- 
eter, Fig.  100,  it  is  possible  to  detect 
very  slight  movements,  and  to  meas- 
ure differences  of  potential  less  than 
0.0001  volt. 

The  capillary  electrometer  is  an  ex- 
cellent "null"  instrument.  In  using 
the  electrometer  no  large  electromo- 
tive force  should  be  applied,  since 
the  meniscus  surface  becomes  polar- 
ized very  easily.  If  this  should  occur, 
a  new  surface  may  be  secured  by 
blowing  gently  at  B  and  forcing  a 
drop  of  mercury  out  of  the  capillary 
into  the  bulb.  Lippmann  studied  the 
effect  of  steadily  increasing  potentials 
on  the  movement  of  the  meniscus. 
Plotting  movements  of  the  meniscus 
on  the  axis  of  ordinates,  and  potentials 

on  the  axis  of  abscissae,  he  found  that  there  is  a  maximum  in  the 
curve  corresponding  to  about  0.8  volt.  This  is  the  electromotive 
force  which  must  be  applied  in  order  to  counterbalance  the  differ- 
ence of  potential  produced  by  the  contact  of  dilute  sulphuric  acid 


Fig.  100. 


ELECTROMOTIVE   FORCE 


461 


Sulphuric  Acid- 


Mer.curj/5. 


Fig.  101. 


with  the  surface  of  the  mercury.     At  the  meniscus  surface  an 
electrical   double   layer   is  formed.     The  mercury   is  positively 

charged,  and  above  it  there  must  be  a 
layer  of  negatively  charged  ions,  as 
shown  in  Fig.  101.  Just  how  this  double 
layer  is  formed  is  not  known  with  cer- 
tainty, but  it  has  been  suggested  that 
the  slight  film  of  oxide  which  is  prob- 
ably present  on  the  surface  of  the  pur- 
est mercury,  dissolves  in  the  sulphuric 
acid  forming  a  solution  of  mercurous 
sulphate,  and  from  this  solution  the 
positively  charged  Hg"  ions  deposit  on 
the  mercury,  giving  it  a  positive  charge. 
Whether  this  explanation  is  correct  or 
not,  the  fact  remains  that  the  mercury 
is  positive  against  the  solution. 

Normal  Electrodes.  The  method  commonly  employed  for 
the  measurement  of  the  difference  of  potential  between  a  metal 
and  a  solution,  is  based  upon  the  use  of  an  electrode  in  which  the 
difference  of  potential  between  the  electrode  and  a  certain  solution 
of  one  of  its  salts  is  known.  Such  an  electrode  is  called  a  normal 
electrode.  If  a  cell  is  made  up  by  combining  the  normal  electrode 
with  the  electrode  whose  potential  is  to  be  determined,  it  is  possible, 
from  measurements  of  the  resulting  electromotive  force,  to  cal- 
culate the  value  of  the  unknown  difference  of  potential.  The  most 
convenient  electrode  to  prepare  is  the  normal  calomel  electrode,  a 
satisfactory  form  of  which  is  shown  in  Fig.  102.  The  bottom  of 
the  electrode  vessel  is  covered  with  a  layer  of  pure  mercury,  upon 
which  is  poured  a  paste,  prepared  by  rubbing  together  in  a  mortar 
mercury  and  calomel,  moistened  with  a  molar  solution  of  potassium 
chloride.  The  vessel  is  ihen  filled  with  a  molar  solution  of  the 
same  salt  which  has  been  saturated  with  calomel  by  prolonged 
shaking  with  the  latter.  Connection  with  the  mercury  is  estab- 
lished by  means  of  a  platinum  wire  sealed  into  a  glass  tube  A ,  the 
latter  being  passed  through  the  rubber  stopper  which  closes  the 
vessel.  In  using  the  calomel  electrode,  the  bent  side  tube  C  is 


462 


THEORETICAL  CHEMISTRY 


filled  with  molar  potassium  chloride  by  applying  suction  at  the 
side  tube  B,  which  is  then  closed  by  means  of  a  pinch-cock. 

The  difference  of  potential,  at  any  temperature  t,  of  the  calo- 
mel electrode  prepared  as  described,  and  represented  by  the 
scheme 

Hg  -  Solution  HgCl  in  molar  KC1, 
is 

TT  =  -0.560  1 1  +  0.0006  (t  -  18)  j  volt. 

The  negative  sign  indicates  that  the  solution  is  negative  to  the 
electrode.     In  order  to  measure  the  potential  of  another  electrode 


Fig.  102. 


Fig.  103. 


by  means  of  the  calomel  electrode,  the  arrangement  shown  in 
Fig.  103  is  commonly  used.  Here  A  represents  the  "  half  -element" 
of  which  the  potential  is  to  be  determined,  B  represents  the  side 
tube  of  the  calomel  electrode,  and  C  represents  an  intermediate, 
connecting  vessel  containing  a  molar  solution  of  potassium  chloride. 
In  cases  where  potassium  chloride  forms  a  precipitate  with  the 
electrolyte  in  A,  the  solution  in  C  may  be  replaced  by  a  molar 


ELECTROMOTIVE  FORCE  463 

solution  of  potassium  nitrate  without  altering  the  value  of  the 
electromotive  force  of  the  cell.  The  original  measurement  of 
the  potential  of  the  calomel  electrode  was  made  by  forming  a 
cell  with  this  and  another  electrode  whose  potential  against  its 
solution  is  zero.  Such  an  electrode  is  known  as  a  null  electrode. 
Thus,  if  a  copper  electrode  is  immersed  in  a  solution  of  copper 
sulphate,  the  Cu"  ions  will  leave  the  solution  and  charge  the 
electrode  positively.  If  now  a  solution  of  potassium  cyanide  is 
added,  the  nearly  undissociated  salt,  K2Cu2  (CN)4,  will  be  formed, 
and  by  adding  a  sufficient  amount  of  the  solution,  the  concentration 
of  the  Cu"  ions  may  be  reduced  until  the  metal  and  the  solution 
have  the  same  potential.  The  addition  of  more  potassium  cyanide 
will  still  further  dimmish  the  osmotic  pressure  of  the  Cu"  ions, 
and  the  electrode  will  acquire  a  negative  charge.  Similarly,  mer- 
cury in  a  solution  of  a  double  cyanide  may  be  used  as  a  null  elec- 
trode. 

Another  form  of  null  electrode  is  the  so-called  dropping  elec- 
trode of  Helmholtz.*  The  principle  involved  "in  this  electrode 
has  already  been  discussed  in  connection  with  Palmaer's  experi- 
ment (p.  453).  An  extremely  fine  stream  of  mercury  is  allowed 
to  flow  from  a  funnel  having  a  minute  capillary  orifice:  the  stem 
of  the  funnel  dips  below  the  surface  of  a  molar  solution  of  potas- 
sium chloride  containing  mercurous  ions.  As  each  little  globule 
enters  the  solution,  it  acquires  a  positive  charge  and  attracts  the 
negatively  charged  ions  of  the  electrolyte,  dragging  them  down 
with  itself.  When  the  globule  reaches  the  layer  of  mercury  at 
the  ^bottom  of  the  vessel,  its  surface  and  capacity  are  diminished, 
and  as  many  Hg*  ions  leave  the  layer  of  mercury  and  enter  the 
solution  as  there  were  negatively  charged  ions  carried  down  by 
the  globule.  This  process  continues  until  the  osmotic  pressure 
of  the  remaining  ions  is  equal  to  the  solution  pressure  of  the  metal: 
the  mercury,  both  in  the  stream  and  at  the  bottom  of  the  vessel, 
has  the  same  potential  as  the  solution.  If  now  the  difference  of 
potential  between  the  mercury  in  the  funnel  and  the  mercury  in 
the  vessel  be  measured,  we  shall  obtain  the  potential  of  mercury 
against  a  molar  solution  of  potassium  chloride.  The  dropping 
*  Ann.  der  Phys.,  44,  42  (1890). 


464  THEORETICAL  CHEMISTRY 

electrode  was  for  a  long  time  regarded  as  an  ideal  standard  of 
potential,  but  Nernst  has  quite  recently  pointed  out  a  number 
of  serious  objections  to  it.  Until  a  wholly  satisfactory  standard 
of  potential  is  obtained,  he  proposes  that  the  potential  of  the 
hydrogen  electrode  be  adopted  as  the  standard.  This  consists 
of  a  strip  of  platinized  platinum,  half  in  pure  hydrogen  gas  and 
half  in  a  solution  of  sulphuric  acid  of  such  concentration  that 
it  shall  contain  1  gram  of  hydrogen  ions  per  liter.  The  use  of 
the  hydrogen  electrode  as  a  standard  is,  of  course,  purely  arbi- 
trary, but  there  are  many  advantages  in  referring  differences  of 
potential  to  this  standard.  Owing  to  certain  experimental  diffi- 
culties attending  the  use  of  this  electrode,  it  is  customary  to 
make  the  actual  measurements  with  the  calomel  electrode,  and 
then  refer  them  to  the  hydrogen  standard,  taking  the  potential 
of  the  calomel  electrode  to  be  —  0.283  volt  when  referred  to 
the  hydrogen  electrode  as  zero.  The  negative  sign  indicates  that 
the  solution  is  negative  against  the  electrode. 

Measurement  of  the  Difference  of  Potential  between  a  Metal 
and  a  Solution.  The  difference  of  potential  between  a  metal 
and  a  solution  of  one  of  its  salts  is  easily  determined  by  means  of 
the  calomel  electrode.  For  example,  in  order  to  determine  the 
potential  of  zinc  against  a  molar  solution  of  zinc  sulphate,  the 
electromotive  force  E,  of  the  combination 

Zn  -  m  ZnSO4 1|  m  KC1,  HgCl  -  Hg, 

(cal.  electrode) 

is  measured  and  found  to  be  1.08  volts,  the  mercury  being  the 
positive  terminal  of  the  cell.  Applying  the  Nernst  equation, 

,-,  RT ,       .Pi      RT ,       PZ 

«.„_„  = —tog.-' ..—log.-, 

in  which  PI  and  p\  denote  the  solution  pressure  and  the  osmotic 
pressure  of  the  zinc  ions,  and  P2  and  p2  denote  the  solution  pressure 
and  the  osmotic  pressure  of  the  mercury  ions,  we  have 

prrr  p 

1.08  =log.-(-  0.56), 


ELECTROMOTIVE  FORCE  465 

or 

D/77  T> 

nUoge  —  =  1.08  -  0.56  =  0.52  volt. 
&  r          pi 

That  is,  the  zinc  electrode  is  negative  against  a  molar  solution  of 
zinc  sulphate,  the  difference  of  potential  being  0.52  volt.  As  an 
example  of  a  cell  in  which  the  mercury  of  the  calomel  electrode 
is  the  negative  terminal  of  the  cell,  we  may  take  the  following 
combination :  — 

Cu  -  m  CuS04 1|  m  KC1,  HgCl  -  Hg. 

The  electromotive  force  of  this  cell  is  0.025  volt.  Since  the 
current  flows  from  the  copper  to  the  mercury,  we  have 

E  =  0.025  =  -0.56  - 
or 


^log,-1  =  -0.025  -  0.56  =  -0.585  volt. 
Z  r          pi 

That  is,  the  copper  electrode  is  positive  against  a  molar  solution 
of  copper  sulphate.  From  the  above  results  it  is  possible  to  cal- 
culate the  electromotive  force  of  the  combination 

Zn  -  m  ZnSO4  [|  m  CuS04  -  Cu. 
Since,  according  to  Nernst's  equation 

RT,      Pi      RT^P* 


where  PI  and  pL  refer  to  the  zinc,  and  P2  and  pz  refer  to  the  copper, 
we  have 

E  =  0.52  -  (-  0.585)  =  1.105  volts. 

Concentration  Elements.  We  now  proceed  to  consider  cells  in 
which  the  electromotive  force  depends  primarily  on  differences 
in  concentration,  —  the  so-called  "concentration  elements." 

Concentration  elements  may  be  conveniently  divided  into  two 
classes:  (a)  elements  in  which  the  electrodes  are  of  different  concen- 
trations, and  (b)  'elements  in  which  the  solutions  are  of  different 
concentrations. 


466 


THEORETICAL  CHEMISTRY 


(a)  Elements  in  which  the  Electrodes  are  of  Different  Concentra- 
tions.    (Amalgams  and  Alloys.)     If  in  the  equation 

RT        P!      RT,      P2 


PI  =  Pz,  as  is  the  case  when  the  ionic  concentrations  of  the  two 
solutions  are  identical,  then  we  have 


where  PI  and  PZ  are  the  respective  solution  pressures  of  the  metal 
dissolved  in  the  electrodes.  If  the  amalgams  are  dilute,  the 
osmotic  pressure  of  the  dissolved  metal  will  be  proportional  to  the 
solution  pressure  of  the  electrode,  and  since  osmotic  pressure 
is  proportional  to  concentration,  we  may  replace  PI  and  P2  in  the 
above  formula  by  the  proportional  terms,  Ci  and  (%,  the  respec- 
tive concentrations  of  the  metal  in  the  two  electrodes.  Hence, 
we  have 

E  =  —lo    - 

The  accuracy  of  this  equation  has  been  fully  established  by  the 
experiments  of  Meyer,*  and  Richards  and  Forbes. f 

Meyer's  results  for  zinc  amalgams  in  solutions  of  zinc  sulphate 
are  given  in  the  accompanying  table. 


A      T' 

degrees. 

Cl 

C2 

E  (obs.). 

E  (calc.). 

284.6 

0.003366 

0.00011305 

0.0419 

0.0416 

291.0 

0.003366 

0.00011305 

0.0433 

0.0425 

285.4 

0.002280 

0.0000608 

0.0474 

0.0445 

333.0 

0.002280 

0.0000608 

0.0520 

0.0519 

The  agreement  between  the  observed  and  calculated  values  of  E 
is  all  that  can  be  desired.  That  the  above  formula  holds  for 
zinc  amalgams  may  be  considered  as  a  proof  of  the  fact  that  the 
zinc  dissolves  in  the  mercury  as  monatomic  molecules.  Thus, 
suppose  the  zinc  to  be  present  in  the  mercury  in  the  form  of  dia- 

*  Zeit.  phys.  Chem.,  7,  477  (1891). 

t  Publication  of  the  Carnegie  Institution,  No.  56. 


ELECTROMOTIVE  FORCE  467 

tomic  molecules;  then  while  the  electrical  energy  would  be  equal 
to  2  FE,  the  osmotic  work  required  to  develop  this  energy  would 

be  ^=  RT  loge  —  ,  hence  we  should  have 
2  c 


or  the  calculated  value  of  the  electromotive  force  would  be  just 
one-half  of  the  observed  value.  The  mercury  in  the  amalgam 
has  been  shown  to  exert  no  effect  upon  the  electromotive  force  of 
the  cell  so  long  as  the  dissolved  metal  has  the  greater  potential. 

(b)  Elements  in  which  the  Solutions  are  of  Different  Concentra- 
tions. In  this  type  of  cell  we  have  two  electrodes  of  the  same 
metal  immersed  in  solutions  of  different  ionic  concentrations  of 
the  metal.  Hence,  we  may  put  PI  =  P2  in  the  equation 

RT,      P1      RT,      P2 

E  =  —F,  loge  —    —   —p,  loge  —  , 

nF     3  pi      nF     &  ip2 
which  then  takes  the  form 


Since  osmotic  pressure  is  proportional  to  concentration,  pi  and 
pz  may  be  replaced  by  the  proportional  terms  Ci  and  02,  and  the 
foregoing  equation  becomes 

,,      RT,      d 

E  =  wl0^' 

or 

RT 


where  mi  and  m%  are  the  molar  concentrations  of  the  two  solutions 
and  cti  and  az  are  the  corresponding  degrees  of  ionization.  As  an 
example  of  a  concentration  element  of  this  class  we  may  take  the 
following  :  — 

Ag  -  0.01  m  AgNO3  1|  0.1  m  AgN03  -  Ag. 


468  THEORETICAL  CHEMISTRY 

The  degrees  of  ionization  of  the  two  solutions  at  18°  are  as  follows: 
-for  0.01  m  AgN03,  a  =  0.93,  and  for  0.1  m  AgNO2,  a  =  0.81. 
Substituting  in  the  equation 

E  =  0.058  log 

° 

we  have 

E  =  0.058  log*0  0.0545  volt. 


The  value  of  E  found  by  direct  experiment  is  0.055  volt. 

In  the  example  just  given,  the  electrodes  are  reversible  with 
respect  to  the  positive  ion  of  the  electrolyte.  Such  electrodes  are 
known  as  electrodes  of  the  first  type.  It  is  also  possible  to  construct 
cells  with  electrodes  which  are  reversible  with  respect  to  the  nega- 
tive ion  of  the  electrolyte.  These  are  termed  electrodes  of  the  second 
type.  The  calomel  electrode  is  an  example  of  an  electrode  of  this 
latter  type.  If  positive  electricity  passes  from  the  metal  to  the 
solution,  the  mercury  combines  with  the  Cl'  ions  forming  mercu- 
rous  chloride,  and  if  positive  electricity  passes  in  the  reverse 
direction,  chlorine  dissolves  and  mercurous  chloride  is  formed. 
In  other  words  the  electrode  behaves  like  a  chlorine  electrode, 
giving  up  or  absorbing  the  element  according  to  the  direction  of 
the  current.  A  typical  combination  involving  an  electrode  of 
the  second  type  is  the  following  :  — 

Ag  -  0.1  m  AgNO3  -  KNO3  -  0.1  m  KC1,  AgCl  -  Ag. 
This  particular  combination  was  studied  by  Goodwin  *  with  a 
view  to  determining  the  solubility  of  silver  chloride.     If  we  assume 
a  saturated  solution  of  silver  chloride  to  be  completely  ionized, 
then  the  solubility  product  will  be 

CAg-   X   CCV  =  S. 

Since  the  concentrations  of  the  two  ions,  Ag"  and  Cl',  are  equal, 
it  follows  that  Vs  will  be  equal  to  the  solubility  of  the  silver 
chloride.  The  electromotive  force  of  a  concentration  cell  at  25° 
is  given  by  the  equation 

,-,      0.0595  . 
log 


n 
*  Zeit.  phys.  Chem.,  13,  577  (1894). 


ELECTROMOTIVE  FORCE  469 

or 

En 


00595 

The  value  of  E  at  25°  for  the  above  cell  was  found  to  be  0.450  volt. 
The  degrees  of  ionization  of  the  two  electrolytes  are  as  follows  :  — 
for  0.1  molar  AgNO3,  a  =  0.82,  and  0.1  molar  KC1,  a  =  0.85. 
Substituting  in  the  preceding  equation,  we  obtain 

0.1  X  0.82       0.450 

~^      =  0^595' 
therefore, 

cs  =  2.24  x  10~9. 

Or,  1.94  X  10~9  is  the  concentration  of  the  Ag*  ion  in  mols  per 
liter  in  a  0.1  molar  potassium  chloride  solution  of  silver  chloride. 
Hence  the  solubility  product  s  will  be 

s  =  2.24  X  10-9  X  0.085  =  1.91  X  lO"10, 
and 

Vs  =  1.38  X  10~5; 

that  is,  the  solubility  of  silver  chloride  in  a  saturated  aqueous 
solution  is  1.28  X  10~5  mol  per  liter  at  25°. 

The  Difference  of  Potential  at  the  Junction  of  the  Solutions 
of  Two  Electrolytes.  Thus  far  we  have  not  taken  into  consider- 
ation the  potential  differences  which  may  be  established  at  the 
junction  of  two  solutions.  Nernst  *  has  shown  that  in  many  cases 
it  is  possible  to  calculate  these  differences  of  potential  by  means 
of  his  osmotic  theory  of  the  origin  of  electromotive  force,  and  the 
values  obtained  are  in  close  agreement  with  the  results  of  experi- 
ment. Let  us  imagine  that  two  solutions  of  hydrochloric  acid  of 
different  concentrations  are  brought  together  so  as  to  avoid  mix- 
ing, the  acid  in  each  solution  being  highly  ionized.  The  hydro- 
gen and  chlorine  ions  will  diffuse  independently,  and  since  the 
former  move  with  the  greater  velocity,  the  more  dilute  solution 
will  soon  contain  an  excess  of  H'  ions  and  the  more  concentrated 
solution  an  excess  of  Cl'  ions.  The  more  dilute  solution  will  be- 
come positively  charged  owing  to  the  presence  of  an  excess  of  H" 
*  Zeit.  phys.  Chem.,  4,  129  (1889). 


470  THEORETICAL  CHEMISTRY 

ions,  while  the  more  concentrated  solution  will  acquire  a  negative 
charge  due  to  the  presence  of  an  excess  of  CI'  ions.  The  accumu- 
lation of  positive  electricity,  however,  will  retard  the  velocity  of 
the  H*  ions  and  accelerate  the  velocity  of  the  CY  ions,  so  that 
ultimately  the  two  ions  will  move  with  the  same  velocity.  The 
difference  of  potential  produced  in  this  way  will  cease  to  exist 
when  the  two  solutions  have  acquired  the  same  concentration. 

In  general,  it  may  be  stated  that  the  difference  of  potential  set 
up  at  the  junction  of  the  two  solutions  is  caused  by  the  differ- 
ence in  the  rates  of  migration  of  the  two  ions,  the  more  dilute 
solution  acquiring  a  charge  corresponding  to  that  of  the  faster 
moving  ion. 

Electromotive  Force  of  Concentration  Cells  with  Transference. 
Let  u  and  v  be  the  migration  velocities  of  the  cation  and  the 
anion  respectively,  and  let  p\  be  the  osmotic  pressure  of  the  ions 
in  the  concentrated  solution  and  pz  the  osmotic  pressure  of  the 
ions  in  the  dilute  solution.  Now  let  one  faraday  of  electricity  pass 
through  the  two  solutions,  the  current  entering  on  the  concen- 
trated side;  then  -  -  gram-equivalents  of  positive  ions  will 

migrate  from  the  concentrated  to  the  dilute  solution,  while 


u  +  v 

gram-equivalents  of  negative  ions  will  migrate  from  the  dilute  to 
the  concentrated  solution.  The  maximum  work  done  by  the 
process  will  be 


RT  log,  2-'  -  -L_  RT  log.**  =          RT  log, 
&  &  6 

corres 
have 


,  -  -  -_  .     = 

u  +  v  &p2      u  +  v          &p2      u  +  v 

The  corresponding  electrical  energy  developed  is  irF,  hence  we 


u-v    RT 


or,  since  osmotic  pressure  is  proportional  to  concentration,  we 
may  substitute  Ci  and  c^  for  pi  and  pz  in  the  preceding  equation, 
and  obtain  the  following  expression  for  the  electromotive  force 
at  the  junction  of  the  two  solutions:  - 

u-v    RT 


ELECTROMOTIVE  FORCE  471 

As  an  example  of  the  use  of  the  above  formula,  we  may  take  the 
calculation  of  the  electromotive  force  of  the  following  combina- 
tion :  — 

Ag  -  0.1  m  AgNOs  -  0.01  m  AgN03  -  Ag. 

(a)  (b)  (c) 

Taking  the  junctions  (a),  (b),  and  (c)  in  order,  we  obtain 
E  =  RTl      C_j.u-v    RT*      ci      RT,      C 

2v        RT 
u  +  v  *    F 

2  v 
But  =  na,  the  transport  number  of  the  anion,  and  therefore 

we  may  write  the  preceding  equation  in  the  form 

(1) 

V  01 

Assuming  the  temperature  to  be  17°  and  passing  to  Briggsian 
logarithms,  we  have 

E  =  0.116  na  log-- 

The  transport  number  for  the  anion  of  silver  nitrate  is  0.522, 
while  ci  =  mien  =  0.1  X  0.82  =  0.082,  and  ca  =  m**  =  0.01  X 
0.94  =  0.0094.  Hence, 

E  =  -2  X  0.522  X  0.058  X  ' 

or 

E  =  -0.057  volt. 

The  value  found  by  direct  experiment  is  —0.055  volt. 

,    It  is  to  be  noted  that  if  the  electromotive  force  at  the  junction 

of  the  two  solutions  is  negative,  equation  (1)  takes  the  form 


where  nc  is  the  transport  number  of  the  cation. 

Electromotive  Force  of  Concentration  Cells  without  Trans- 
ference.    A  less  familiar  type  of  concentration  cell  is  that  which 


472  THEORETICAL  CHEMISTRY 

does  not  involve  transference.    'The  following  combination  may 
be  taken  as  an  example  of  such  a  cell  :  — 

Ag|AgCl,KCl|K(Hg),    |  K(Hg),   KC1,  AgCl  |  Ag. 

Ci  C2 

It  will  be  observed  that  this  cell  is  in  reality  made  up  of  two 
independent  cells  and  involves  no  liquid  junction.  While  the 
elimination  of  diffusion  appreciably  simplifies  the  theoretical 
treatment  of  cells  of  this  type,  practically  it  is  difficult  to  find 
electrodes  which  are  reversible  toward  both  ions  of  the  electrolyte. 
The  passage  of  one  faraday  of  electricity  through  the  cell  results 
in  the  formation  on  the  dilute  side  of  one  equivalent  of  potassium 
chloride  from  the  silver  chloride  and  the  amalgam,  while  on  the 
concentrated  side  a  corresponding  amount  of  potassium  chloride 
is  decomposed.  If  we  assume  the  solutions  to  be  so  dilute  as  to 
be  completely  ionized,  one  equivalent  of  potassium  chloride  will 
function  as  two  mols  of  gas,  and  the  electromotive  force  of  the  cell 
will  be 


Formulas  for  the  Difference  of  Potential  at  Liquid  Junctions. 
As  has  been  pointed  out,  the  total  electromotive  force  of  a  con- 
centration cell  with  transference  is  made  up  of  the  algebraic  sum 
of  the  potentials  at  the  two  electrodes  and  the  potential  at  the 
junction  of  the  two  solutions.  Since  the  value  of  the  potential  at 
the  electrodes  alone  is  often  desired,  numerous  formulas  have  been 
derived  for  the  calculation  of  the  potential  at  the  liquid  junction. 

In  addition  to  the  formula  of  Nernst,  already  mentioned,  for- 
mulas have  been  proposed  by  Planck,*  Henderson,  f  Gumming,  j: 
Lewis  and  Sargent,  §  and  Maclnnes.  ||  Of  these  different  formulas 
that  of  Maclnnes  possesses  distinct  advantages. 

*  Wied.  Ann.,  40,  561  (1890). 

f  Zeit.  phys.  Chem.,  59,  118  (1906);  63,  325  (1908). 

j  Trans.  Faraday  Soc.,  8,  86  (1912);  9,  174  (1913). 

§  Jour.  Am.  Chem.  Soc.,  31,  363  (1909). 

||  Ibid.,  37,  2301  (1915). 


ELECTROMOTIVE   FORCE  473 

If  one  faraday  of  electricity  be  passed  through  the  cell 
Ag|AgCl,  KC1|KC1,  AgCl|Ag, 

Cl  Cz 

one  equivalent  of  chloride  ions  will  enter  the  dilute  solution  while 
a  corresponding  amount  will  be  electrolyzed  out  of  the  more  con- 
centrated solution.  The  current  will  be  carried  across  the  liquid 
junction  by  the  migration  of  nc  equivalents  of  potassium  ions  in 
the  direction  of  the  current  and  by  the  migration  of  (1  —  nc) 
equivalents  of  chloride  ions  in  the  opposite  direction.  The  total 
effect  of  the  passage  of  one  faraday  of  electricity  will  be  the  trans- 
ference of  one  equivalent  of  salt  from  the  more  concentrated  to  the 
more  dilute  solution.  The  osmotic  work  at  the  junction  of  the 
two  solutions  will  correspond  to  the  algebraic  sum  of  the  number 
of  ion  equivalents  which  have  migrated  from  the  concentrated  to 
the  dilute  solution,  or  nc  —  (1  —  nc)  =  2  nc  —  1.  ( 

In  order  to  obtain  the  electrical  energy  necessary  to  perform 
this  amount  of  osmotic  work,  let  us  consider  the  following  cell 
involving  no  transference :  — 

Ag  |  AgCl,  KC1 1  K  (Hg)z  ||  K  (Hg)x  |  KC1,  AgCl  |  Ag. 

Ci  C2 

The  passage  of  one  faraday  of  electricity  through  this  cell  in- 
volves, as  we  have  seen,  the  formation  of  one  equivalent  each  of 
potassium  and  chloride  ions  in  the  dilute  solution,  and  the  re- 
moval of  one  equivalent  of  each  ion  from  the  more  concentrated 
solution.  The  electrical  energy  accompanying  the  transfer  of  two 
ion  equivalents  from  one  solution  to  the  other  will  be  equal  to 
the  product  of  the  electromotive  force  of  the  cell  and  one  faraday, 
or  EF. 

The  electromotive  force  IT,  at  the  liquid  junction,  can  now  be 
obtained  by  the  simple  proportion 

EF  :  irF  =  2  :  2  nc  -  1, 

.  =  g^-1)-.  CD 

Since  the  ratio  of  E()  the  electromotive  force  of  the  cell  with  trans- 
ference, to  E,  the  electromotive  force  of  the  cell  without  trans- 


474  THEORETICAL  CHEMISTRY 

ference,  is  equal  to  nc,  it  follows  that  equation  (1)  may  be  written 
in  the  form 

(2) 


It  is  to  be  observed  that  equations  (1)  and  (2)  involve  no  assump- 
tions concerning  the  concentrations  of  the  ions  in  the  solutions,  one 
of  the  characteristics  which  distinguishes  these  formulas  from  any 
of  the  others  hitherto  proposed. 

Lewis  and  Sargent  *  have  proposed  a  modification  of  a  formula 
derived  by  Planck  for  the  difference  of  potential  at  the  junction  of 
two  equally  concentrated  electrolytes  having  a  common  ion.  If 
AI  and  A2  are  the  equivalent  conductances  of  the  two  electrolytes, 
the  value  of  the  liquid  junction  potential  TT  is  given  by  the  formula 


The  junction  between  two  different  concentrations  of  two  elec- 
trolytes with  a  common  ion  may  be  replaced  by  two  junctions, 
one  of  which  connects  two  different  concentrations  of  the  same 
electrolyte,  and  the  other  connects  two  different  electrolytes  of  the 
same  concentration.  For  example,  the  junction 

0.1NNaCl|0.05NHCl 
may  be  replaced  by 

0.1  N  NaCl  |  0.05  N  NaCl  |  0.05  N  HC1 

(A)  (B) 

in  which  the  potential  of  the  junction  (A)  may  be  calculated  by 
equations  (1)  or  (2)  and  the  potential  of  junction  (B)  by  equa- 
tion (3). 

The  electromotive  force  at  the  junction  of  solutions  of  differently 
concentrated  uni-univalent  ionogens  which  do  not  contain  a  com- 
mon ion  may,  in  like  manner,  be  calculated  by  employing  three 
junctions.  Thus,  the  junction 

0.1  N  KN03  1  0.05  N  NaCl 
*  Loc.  cit. 


ELECTROMOTIVE  FORCE  475 

may  be  replaced  by 

0.05  N  NaCl  |  0.1  N  NaCl  |  0.1  N  KC1  1  0.1  N  KNO3. 

(A)  (B)  (BO 

The  potential  at  (A)  may  be  calculated  by  means  of  equations  (1) 
or  (2)  and  the  potentials  at  (B)  and  (B')  by  means  of  equation  (3). 

By  the  use  of  equations  (1),  (2),  and  (3)  in  the  manner  indi- 
cated, it  is  possible  to  calculate  the  junction-potential  between  the 
solutions  of  any  two  uni-univalent  ionogens  to  within  a  few  tenths 
of  a  millivolt. 

Normal  Electrode  Potential.  The  difference  of  potential  at 
a  reversible  electrode,  when  the  concentration  of  the  ions  of  the 
electrolyte  is  normal,  is  known  as  the  normal  electrode  potential. 

According  to  Nernst,  the  expression  for  the  difference  of  potential 
at  a  single  electrode  is 

RT 


RT 


where  P  is  the  solution  pressure  of  the  electrode,  p  is  the  osmotic 
pressure  of  the  ions  of  the  electrolyte,  and  where  C  and  c  are  con- 
centrations proportional  to  P  and  p  respectively.  If  the  concen- 
tration of  the  ions  is  unity,  c  =  1,  and  the  expression  for  the 
normal  electrode  potential  E0  becomes 


(2) 

On  subtracting  (1)  from  (2),  we  obtain 

(3) 


where  c  is  the  concentration  of  the  ions  in  a  solution  whose  elec- 
trode potential  is  TT.  In  the  actual  determination  of  normal  elec- 
trode potentials,  the  value  of  c  in  equation  (3)  is  usually  very  much 
less  than  unity. 

The  following  table  gives  the  most  important  electrode  poten- 
tials, the  data  in  the  second  column  being  referred  to  the  normal 


476 


THEORETICAL   CHEMISTRY 


calomel  electrode  as  zero,  and  the  data  of  the  third  column  being 
referred  to  the  normal  hydrogen  electrode  as  zero. 

NORMAL  ELECTRODE  POTENTIALS. 


Electrode. 

N.  E.  Potential, 
N.  Calomel  Electrode 
.          =°- 

\.  H.  Potential, 
N.  Hydrogen  Electrode 
=  0. 

Cations. 


Copper  

+0.046 

+0.329 

Hydrogen 

—0  283 

+0  000 

Lead 

—0  412 

-0  129 

Mercury                         .    . 

+0  467 

+0  750 

Platinum            

+0.580  oa. 

+0  863  ca. 

Potassium           

-3.48 

-3  20 

Silver  
Sodium    

+0.515 
+2.998 

+0.718 

+3.281 

Tin   

-0.475 

-0.192 

Zinc      

-1.053 

-0.770 

Anions. 


Bromine  

+0.797 

+  1.08 

Chlorine 

+  1.067 

+1  35 

Iodine 

+0.345 

+0  628 

Oxygen 

+0.110 

+0  393 

It  should  be  observed  that  the  value  of  the  normal  electrode 
potential  cannot  be  calculated  directly  by  means  of  equation  (2) 
owing  to  the  uncertainty  attaching  to  the  quantity  C.  In  fact  the 
best  interpretation  that  we  have  of  the  physical  significance  of  C 
is,  that  it  is  a  quantity  whose  logarithm  is  proportional  to  the 
normal  electrode  potential. 

Electrometric  Determination  of  Valence.  The  equation  of 
Nernst  for  the  electromotive  force  of  a  concentration  cell  may  be 
employed  to  determine  the  valence  of  the  ions. 

For  example,  the  chemical  behavior  of  mercurous  salts  is  such 
as  to  justify  the  use  of  single  or  double  formulas  involving  either 
Hg*  or  Hg2*.  To  determine  which  of  these  two  formulas  is  correct, 
Ogg  *  set  up  the  following  cell :  — 

Hg  |  0.5  N  Mercurous  Nitrate  ||  0.05  N  Mercurous  Nitrate  |  Hg 
0.1  N  HN03  0.1  N  HN03 

*  Zeit.  phys.  Chem.,  27,  285  (1898). 


ELECTROMOTIVE  FORCE  477 

and  found  its  electromotive  force  at  17°  to  be  0.029  volt.  If  we 
neglect  the  difference  of  potential  at  the  junction  of  the  two  solu- 
tions, the  electromotive  force  of  this  cell  may  be  calculated  by  the 
familiar  formula 


Assuming  that  c^/Ci  is  equal  to  10,  and  passing  to  Briggsian  loga- 
rithms, we  have 

0.029  =  0.058/n, 
or  n  =  2. 

Hence  the  valence  of  the  mercurous  ion  is  2  and  it  must,  in  con- 
sequence, be  represented  by  Hg2*.  It  follows  that  the  correct  for- 
mula of  mercurous  nitrate  is  Hg2(NO3)2. 

Electronic  trie  Determination  of  Transport  Numbers.  The 
formula  for  the  electromotive  force  of  a  concentration  cell  with 
transference  has  been  shown  to  be 

«> 


If  the  electromotive  force  of  such  a  combination  is  measured  and 
the  values  of  Ci  and  &  are  accurately  known,  obviously  the  value 
of  the  transport  number  of  the  cation  nc,  can  be  calculated.  If, 
after  having  measured  Et)  a  similar  cell  without  transference  be 
set  up  and  its  electromotive  force  measured,  it  is  possible  to  obtain 
an  expression  for  nc  which  does  not  involve  either  Ci  or  c%.  As  has 
been  shown,  the  electromotive  force  of  a  cell  without  transference, 
E,  is  given  by  the  formula 

« 


Dividing  equation  (1)  by  equation  (2),  we  obtain 


which  expression  gives  the  transport  number  in  terms  of  the  two 
measured  electromotive  forces.     This  method  for  the  determina- 
tion of  transport  numbers  was  first  suggested  by  Helmholtz.* 
The  transport  number  of  the  lithium  ion  has  recently  been  de- 
*  Ges.  Abhl.  I,  840:  II,  979. 


478 


THEORETICAL  CHEMISTRY 


termined  in  this  manner  by  Pearce  and  Mortimer  *  using  cells  made 
up  according  to  the  following  schemes  :  — 

Ag  |  AgCl,  LiCl  ||  LiCl,  AgCl  |  Ag     (with  transference), 

Ci  Cz 

Ag  |  AgCl,  LiCl  |  Li  (Hg)x  1  1  Li  (Hg)x  |  LiCl,  AgCl  |  Ag  /     without    \ 
ci  ^2  Vtransference/ 

A  comparison  between  their  results  and  the  values  given  by 
Kohlrausch  and  Holborn  f  is  afforded  by  the  accompanying  table. 

Concentration  Ratio  ......   1.0-0.1     0.5-0.05    0.1-0.01     0.05-0.005 

Mean  nc  (K.  and  H.)  .....       0.285          0.300          0.340  0.360 

Mean  nc  (P.  and  M.)  .....       0.279          0.322          0.343  0.365 

It  will  be  observed  that  the  agreement  between  the  two  series  of 
results  is  satisfactory. 

Concentration  and  Activity.  If  in  the  equation  for  the  elec- 
tromotive force  of  a  concentration  cell  without  transference,  viz., 

„  2RT 


we  substitute  the  observed  value  of  E  and  solve  the  equation  for 
the  ratio,  GZ/CI,  we  find  that  the  value  thus  obtained  does  not  agree 
with  the  value  derived  from  conductance  measurements.  In  other 
words,  if  mi  and  mz  are  the  actual  molar  concentrations  of  the  two 
solutions  forming  the  cell,  and  c*i  and  <*2  are  the  corresponding  de- 
grees of  ionization,  we  find  that  the  ratio  mzaz/miai  is  not  equal 
to  the  ratio  GZ/CI  as  calculated  from  the  electromotive  force  of 
the  cell.  This  is  shown  by  the  following  table  which  gives  the 
ratios  of  the  ionic  concentrations  of  solutions  of  potassium  chloride 
corresponding  to  various  ten-to-one  salt  concentrations. 

RATIOS  OF  ION  CONCENTRATIONS  IN  SOLUTIONS  OF 
POTASSIUM  CHLORIDE. 


Concentration 
Ratio. 

C2/C, 

(Conductance). 

C2/Ci 

(Electromotive 
Force). 

0.5    :0.05 
0.1    :0.01 
0.05  :  0.005 
0.01  :  0.001 

8.85 
9.16 
9.30 
9.62 

8.09 

8.33 
8.64 
9.04 

*  Jour.  Am.  Chem.  Soc.,  40,  518  (1918). 
t  Leitvermogen  der  Elektrolyte,  p.  201. 


ELECTROMOTIVE  FORCE 


479 


If  the  values  of  c^/Ci  in  the  table  are  plotted  as  ordinates  against 
the  higher  concentration  of  each  pair  of  solutions  as  abscissae,  we 
obtain  the  curves  shown  in  Fig.  104.  It  will  be  seen  that  notwith- 
standing the  fact  that  the  ratios  differ  widely  when  calculated  by 
the  two  methods,  both  sets  of  figures  approach  the  limiting  value, 
10,  at  infinite  dilution.  From  this  it  is  evident  that  the  equation 


10.0 


0.4 


0.5 


0.2    '  0.3 

Concentration 

Fig.  104. 

of  Nernst  is  only  applicable  to  cells  involving  solutions  which  are 
practically  infinitely  dilute. 

The  figures  in  the  last  column  of  the  table  are  known  as  activity 
ratios,  and  the  individual  values  of  c\  and  GZ  are  termed  activi- 
ties. If  it  be  assumed  that  in  0.001  KC1  the  activity  of  the  ions 
is  identical  with  their  concentration,  as  obtained  by  multiplying 
the  total  concentration  of  the  salt  by  the  ionization  as  determined 
from  the  conductance  ratio,  then  the  activities  in  0.01  N  KC1  and 
0.1  N  KC1  can  be  calculated.  The  values  so  obtained  are  given  in 
the  following  table  together  with  the  corresponding  values  of  the 
ionization  calculated,  (1)  from  the  conductance  ratio,  and  (2)  from 
the  so-called  "  thermodynamically  effective  "  ionization.  this  latter 
being  the  quotient  obtained  by  dividing  the  activity  by  the  corre- 
sponding total  concentration. 

That  the  degree  of  ionization  determined  from  conductance 
measurements  does  not  agree  with  the  value  obtained  from  meas- 
urements of  electromotive  force  has  been  known  for  some  time. 


480 


THEORETICAL  CHEMISTRY 


ACTIVITY  AND  IONIZATION  OF  POTASSIUM  CHLORIDE 
SOLUTIONS. 


Salt  Concentration. 

Activity. 

A/AO. 

"  Thermodynamically 
Effective  "  lonization. 

0.001  N 
0.01    N 
0.1     N 

0.000979 
0.00885 
0.0738 

97.9 
94.1 
86.1 

(97.9) 
88.5 
73.8 

It  is  thought  that  the  decrease  in  activity  of  the  ions  with  increas- 
ing concentration  is  due  to  a  change  in  the  nature  of  the  solvent 
brought  about  by  the  electric  charges  on  the  ions. 

Electrometric  Determination  of  Hydrolysis.  One  of  the  most 
satisfactory  methods  which  we  possess  for  the  determination  of  the 
degree  of  hydrolysis  of  salts  depends  upon  the  measurement  of  the 
electromotive  force  of  cells  made  up  as  follows :  — 

PtH2 1  salt  solution  |  sat.  NH4NO3  *  ||  N  KC1,  H&Cl,  |  Hg. 

From  the  measured  electromotive  force  of  the  cell,  the  concentra- 
tion of  the  hydrogen  ion  in  the  solution  can  be  determined,  and 
from  this  the  degree  of  hydrolytic  dissociation  of  the  salt  can  be 
calculated.  The  method  is  especially  valuable  in  cases  where  the 
concentration  of  the  hydrogen  ion  is  very  small.  Unfortunately 
the  application  of  the  method  is  restricted  to  the  salts  of  metals  less 
noble  than  hydrogen.  In  other  words,  it  cannot  be  used  to  deter- 
mine the  hydrolysis  of  the  salts  of  metals  which  would  be  precipi- 
tated upon  the  platinum  electrode.  The  applicability .  of  the 
method  is  further  limited  by  the  fact  that  certain  ions,  such  as  Fe"", 
N(V,  and  C1CV,  are  either  wholly  or  partially  reduced  by  the 
hydrogen  of  the  hydrogen  electrode. 

The  method  has  been  successfully  applied  by  Denham  f  to 
the  determination  of  the  hydrolysis  of  various  salts,  among  which 
may  be  mentioned,  aluminium  chloride,  aluminium  sulphate,  nickel 
chloride,  nickel  sulphate,  cobalt  sulphate,  and  ammonium  chloride. 

As  an  illustration  of  the  method  we  may  take  the  case  of  am- 

*  A  saturated  solution  of  ammonium  nitrate  is  interposed  between  the 
solutions  to  eliminate  any  difference  of  potential  at  the  junction  of  the  solu- 
tions. 

t  Jour.  Chem.  Soc.,  93,  41  (1908);  Zeit.  anorg.  Chem.,  57,  361  (1908). 


ELECTROMOTIVE   FORCE  481 

monium  chloride  which  dissociates  hydrolytically  according  to 
the  equation 

NH4C1  +  H.OH  <=»  NH4OH  +  HC1. 

Here  we  have  a  weak  base  and  a  very  strong  acid  as  the  products 
of  hydrolytic  dissociation.  While  the  ammonium  hydroxide  may 
be  regarded  as  practically  non-ionized,  the  hydrochloric  acid  is  to 
be  considered  as  having  undergone  complete  ionization.  If  one 
mol  of  ammonium  chloride  is  dissolved  in  v  liters  of  water  and  the 
degree  of  hydrolytic  dissociation  is  x,  then  the  concentrations  of 
the  products  of  the  reaction,  ammonium  hydroxide  and  hydro- 
chloric acid,  will  be  x/v.  Since  the  acid  is  completely  ionized,  x/v 
will  also  represent  the  concentrations  of  the  hydrogen  and  chloride 
ions.  Assuming  that  the  active  mass  of  the  water  remains  con- 
stant, we  have,  according  to  the  law  of  mass  action, 


where  KH  is  the  hydrolytic  constant.  Since  v  is  known,  it  only  re- 
mains to  determine  x/v,  or  the  concentration  of  the  hydrogen  ions, 
in  order  to  be  able  to  calculate  Kh.  The  potential  at  the  hydrogen 
electrode  is  given  by  the  familiar  formula 

RT,       CH- 


or  denoting  the  normal  electrode  potential  R  T  loge  C  by  irQ,  we 
may  write 

.  RT,      x 


According  to  Denham,  the  electromotive  force  of  the  cell, 

PtH2  1  N/32  NH4C1  1  sat.  NH4NO3  1|  N  KC1,  Hg2Cl2  1  Hg, 
at  25°  is  0.6056  volt,  the  cu  rent  flowing  outside  the  cell  in  the 
direction  of  the  arrow.  The  potential  of  the  normal  calomel 
electrode  being  +  0.56  volt,  it  follows  that  the  potential  of  the 
hydrogen  electrode,  TT  =  0.56  -  0.6056  =  -  0.0456  volt.  There- 
fore, 

PT        x 
-  0.0456  =  *o  +^p  log,  -• 


482  THEORETICAL  CHEMISTRY 

The  absolute  value  of  the  potential  of  the  normal  hydrogen  elec- 
trode 7r0,  referred  to  the  normal  calomel  electrode  as  +  0.56  volt 
and  not  as  zero,  is  +  0.277  volt.  Substituting  this  value  in  the 
foregoing  equation  and  passing  to  Briggsian  logarithms,  we  have 

0.059  log-  =  0.0456  -  0.277  =  -  0.2314. 

Solving  this  equation  we  find  x/v  =  0.340  X  106  gram-ions  of 
hydrogen  per  liter.  Since  the  value  of  x/v  would  be  ^  if  the  salt 
were  completely  hydrolyzed,  the  percentage  of  hydrolysis  under 
the  conditions  of  the  experiment,  i.e.,  when  v  =  32,  is 

0.340  x  W  X  100 

The  value  of  the  hydrolytic  constant  is  given  by  the  equation 


.         (0.340  X 
Kh  -  32(1-  0.340  X10-)- 

The  degree  of  hydrolytic  dissociation  of  ammonium  chloride  has 
been  determined  by  Noyes  from  measurements  of  electrical  con- 
ductance. The  value  of  x  for  0.01  N  NH4C1  was  found  by  him  to 
be  0.02  at  18°,  while  Denham  found  x  =  0.018  at  25°  by  extrapola- 
tion of  his  electrometric  data. 

Oxidation  and  Reduction  Elements.  When  a  dissolved  sub- 
stance passes  from  a  lower  to  a  higher  state  of  oxidation,  the 
change  in  the  positive  ion  may  be  considered  as  due  to  an  increase 
in  the  number  of  electrical  charges  on  the  ion;  thus,  when  a  ferrous 
salt  is  oxidized  to  the  ferric  state,  the  change  may  be  represented 
by  the  equation 

Fe"-K(+)-»Fe~ 

Similarly,  the  reduction  of  a  ferric  salt  to  the  corresponding  ferrous 
salt  may  be  represented  by  the  reverse  equation,  or 


The  formation  of  zinc  ions  from  metallic  zinc  may  be  considered 
as  an  oxidation,  and  may  be  represented  by  the  equation 

Zn  +  2  (+)  -*  Zn~. 


ELECTROMOTIVE  FORCE  483 

The  formation  of  negative  ions  from  a  non-metallic  element  may 
be  considered  as  a  reduction,  as  for  example,  the  change  of  potas- 
sium ferricyanide  to  potassium  ferrocyanide,  which  may  be  rep- 
resented by  the  equation 

Fe(CN)6'"  +  (-)  -»Fe(CN)6"". 

The  foregoing  considerations  lead  to  the  following  definition  of 
the  terms  oxidation  and  reduction:  Oxidation  is  the  process  in 
which  a  substance  takes  up  positive  or  parts  with  negative  chargest 
and  reduction  is  the  process  in  which  a  substance  takes  up  negative 
or  parts  with  positive  charges. 

The  potential  difference  between  a  metal  and  a  solution  of  one 
of  its  salts  is  a  measure  of  the  tendency  of  the  metal  to  form  ions, 
or  in  other  words  is  the  criterion  of  its  tendency  towards  oxidation. 
The  tendency  of  an  ion  in  a  lower  state  of  oxidation  to  pass  over 
into  a  higher  state  of  oxidation  may  be  determined  by  means  of 
electromotive  force  measurements. 

A  typical  oxidation  cell  is  the  following:  — 

Pt  -  Fe"  ||  Fe*"  -  Pt. 

If  the  platinum  electrodes  are  connected  to  a  lead  accumulator, 
and  a  current  is  passed  in  the  direction  of  the  arrow 

Pt  -  Fe"  ||  Fe""  -  Pt, 


the  ferrous  ions  on  one  side  of  the  element  will  all  be  oxidized  to 
the  ferric  state,  while  the  ferric  ions  on  the  other  side  will  all  be 
reduced  to  the  ferrous  state,  thus: 

Pt  -  Fe'"  ||  Fe"  -  Pt. 

If  now  the  connection  with  the  accumulator  is  broken  and  the  two 
platinum  electrodes  are  connected  with  a  wire,  the  following  proc- 
ess will  take  place:  —  ferric  ions  will  give  up  positive  charges  to 
one  electrode,  thereby  being  reduced  to  ferrous  ions,  while  the 
charges  given  up  pass  along  the  wire  to  the  other  electrode,  there 
converting  some  ferrous  ions  to  ferric  ions.  This  process  will 
continue  until  an  equilibrium  is  established,  in  which  the  ratio 


484  THEORETICAL  CHEMISTRY 

Fe"  :  Fe*"  will  be  the  same  in  both  solutions.  The  electromotive 
force  corresponding  to  this  change  gives  a  measure  of  the  tendency 
of  the  ions  to  undergo  oxidation,  and  is  directly  proportional  to 
the  ionic  concentrations. 

The  following  modification  of  the  Nernst  equation  .gives  the 
relationship  between  ionic  concentrations  and  the  resulting  elec- 
tromotive force: — 


where  E(CO->C.)  is  the  electromotive  force  produced  by  the  passage 
from  the  lower,  or  "ous  "  state  of  oxidation  to  the  higher,  or  "ic  " 
state  of  oxidation;  P  is  the  potential  when  the  concentrations  of 
the  "ous  "  and  "ic  "  ions  are  equal,  c0  and  d  are  the  ionic  con- 
centrations of  the  lower  and  higher  states  of  oxidation  respectively, 
and  n  is  the  difference  in  valence  of  the  two  kinds  of  ions.  A 
number  of  oxidation  and  reduction  elements  have  been  studied 
by  Bancroft.* 

Heat  of  lonization.  If  the  difference  of  potential  between  a 
metal  and  the  solution  of  one  of  its  salts  is  known,  together  with 
its  temperature  coefficient,  it  is  possible  to  calculate  the  heat  of 
ionization  of  the  metal  by  means  of  the  Gibbs-Helmholtz  equation, 

*  =  nF+TdT' 

Solving  the  equation  for  Q,  the  heat  evolved  when  one  mol  of  ions 
is  formed  at  the  electrode,  we  have 

dir 


For  example,  the  potential  of  zinc  against  a  molar  solution  of  zinc 
chloride  is  —  0.497  volt  at  25°,  and  the  temperature  coefficient  of 
electromotive  force  is  0.000664  volt  per  degree.  Substituting  in 
the  equation,  we  have 

Q  =  [-  0.497  -  (273  +  25)  X  0.000664]  (2  X  96,540  X  0.2394) 

or  Q  =  32,120  calories. 

*  Zeit.  phys.  Chem.,  10,  387  (1892). 


ELECTROMOTIVE  FORCE 


485 


That  is,  the  heat  of  the  reaction 

Zn  +  2  (+)  ->  ZiT 

is  32,120  calories  per  mol  of  zinc. 

Gas  Cells.  It  is  interesting  to  note  that  gases  may  function 
as  electrodes  in  much  the  same  way  as  metals  or  amalgams.  Gas 
electrodes  are  usually  prepared  by  partially 
immersing  strips  of  platinized  platinum  in 
a  solution  of  a  suitable  electrolyte,  and 
bubbling  the  gas  through  the  solution  until 
a  constant  difference  of  potential  is  estab- 
lished between  it  and  the  electrode.  A  very 
satisfactory  form  of  gas  electrode  is  shown  in 
Fig.  105.  Reference  has  already  been  made 
to  the  hydrogen  electrode  in  connection 
with  the  measurement  of  single  electrode 
potentials. 

This  electrode  is  completely  reversible 
and  behaves  like  a  plate  of  metallic  hydro- 
gen, the  reaction  at  the  electrode  being 
represented  by  the  equation 


The  amount  of  energy  developed  by  the 

passage  of  a  certain  quantity  of  gas  into 

the  ionic  state  is  precisely  the  quantity  nec- 

essary and  sufficient  to  produce  the  reverse  Fi8-  105- 

action. 

This  being  true,  the  metal  of  the  electrode  can  exert  no  influence 
upon  the  electromotive  force.  A  hydrogen  concentration  cell 
can  be  formed  by  connecting  two  hydrogen  electrodes,  containing 
the  gas  at  different  pressures,  through  an  intermediate  electrolyte. 
The  direction  of  the  current  is  such  that  the  pressures  on  the  two 
sides  of  the  cell  tend  to  become  equal,  molecular  hydrogen  being 
ionized  on  the  high  pressure  side,  and  ionized  hydrogen  being  dis- 
charged on  the  low  pressure  side. 

The  electromotive  force  of  such  a  cell  can  be  calculated  by  means 


486  THEORETICAL  CHEMISTRY 

of  the  Nernst  equation.  Let  us  consider  a  cell  composed  of  two 
hydrogen  electrodes,  each  at  atmospheric  pressure,  the  H*  ion 
concentration  in  each  being  ci,  then 

f.  til    ,  G!  fll 


where  C\  is  the  molecular  concentration  of  the  hydrogen  dissolved 
in  the  platinum  at  atmospheric  pressure  p\.  Since  the  hydrogen 
is  present  in  the  form  of  diatomic  molecules,  n  =  2. 

If  now  the  pressure  of  the  gas  at  one  electrode  be  increased  to 
pz,  and  the  corresponding  molecular  concentration  of  the  hydro- 
gen in  the  electrode  be  C2,  then  we  shall  have 

T-.          £1,1   i          GI          til.    i        L>2 


or  snce 

Ci  :  C2  ::  p\  :  pz, 

*  -??"*£•  « 

Equation  (1)  applies  equally  well  to  cells  in  which  two  different 
gases  are  employed.  If  solution  pressures  be  used  in  the  calcu- 
lation of  the  electromotive  force  of  a  gas  cell,  equation  (1)  becomes 

geg,  (2) 

where  PI  and  P2  are  the  solution  pressures  of  the  two  gases.  Since 
the  values  of  E  obtained  by  equations  (1)  and  (2)  must  be  equal, 
we  may  write 

RT,      Pl      RT,      Pi 

and 


therefore 


That  is,  the  ratio  of  the  actual  gas  pressures  is  equal  to  the  ratio 
of  the  squares  of  the  corresponding  solution  pressures. 


ELECTROMOTIVE  FORCE  487 

lonization  of  Water.  An  important  application  of  the  gas 
cell  is  its  use  in  determining  the  degree  of  ionization  of  water. 
If  we  measure  the  electromotive  force  of  the  cell 

PtH2  -  0.1  m  NaOH  -  0.1  m  HC1  -  Ptn2, 

and  determine  the  concentration  of  the  H"  ions  on  one  side  and  the 
concentration  of  the  OH'  ions  on  the  other  side,  we  can  calculate 
the  concentration  of  the  H*  ions  in  the  sodium  hydroxide  solution. 
The  reaction  which  produces  the  current  is  represented  by  the 
equation 

NaOH  +  HC1  =  NaCl  +  H2O, 

or  more  correctly 

H-  +  OH'  =  H2O. 

At  25°  the  electromotive  force  of  the  above  cell  is  0.646  volt. 
At  the  junction  of  the  two  solutions  an  electromotive  force  of 
0.0468  volt  is  set  up;  hence  the  true  electromotive  force  of  the 
cell  is  0.646  +  0.0468  =  0.6928  volt.  The  degree  of  ionization 
of  a  0.1  molar  solution  of  hydrochloric  acid  is  «i  =  0.924,  and 
the  degree  of  ionization  of  a  0.1  molar  solution  of  sodium 
hydroxide  is  az  =  0.847.  Introducing  these  values  into  the 
Nernst  equation 


we  have 

0.6928  =  0.0595  log0'1  X0-924. 

C2 

Solving  this  equation  for  C2,  the  concentration  of  the  H*  ions 
in  the  sodium  hydroxide  solution,  we  find  Cz  to  be  equal  to 
1.66  X  10~13.  Therefore 

s  =  CH-  X  COR'  =  1.66  X  10-13  X  0.1  X  0.847  =  1.406  X  lO"14, 
and 

CH-  =  COH'  =  V7  =  Vl.406  X  10~14  =  1.187  X  10~7. 

This  value  is  in  excellent  agreement  with  the  values  obtained  from 
measurements  of  the  electrical  conductance  of  water  and  the  speed 


488  THEORETICAL  CHEMISTRY 

of  hydrolysis  of  esters.  The  values  of  the  degree  of  ionization  of 
water  at  25°,  as  determined  by  these  three  methods,  are  as  fol- 
lows :  — 

Electrical  conductance  of  pure  water 1.05  X  10~7 

Velocity  of  hydrolysis  of  methyl  acetate 1.2    X  10~7 

E.M.F.  of  hydrogen-oxygen  cell 1.18  X  10"7 

When  we  consider  the  exceedingly  small  extent  to  which  water 
is  ionized,  the  close  agreement  between  these  results  is  most  satis- 
factory. The  correctness  of  these  figures  can  be  further  checked 
by  taking  the  values  of  the  degree  of  ionization  of  water  at  two 
temperatures,  and  calculating  the  heat  of  the  reaction 

H'  +  OH'  =  H20, 

by  means  of  van't  HofPs  isochore  equation. 

Thus  according  to  Kohlrausch,  the  degree  of  ionization  of  water 
is  0.35  X  10~7  at  0°,  and  2.48  X  10~7  at  50°.  Introducing  these 
values  into  the  equation 

RX  2.3026  (log  K2- 


and  solving  for  q,  we  obtain  13,810  calories,  a  value  agreeing  well 
with  that  found  by  the  direct  measurement  of  the  heat  of  neu- 
tralization of  completely  ionized  acids  and  bases,  viz.,  13,700 
calories. 

Storage  Cells  or  Accumulators.  Storage  cells  or  accumulators, 
as  the  name  implies,  are  devices  for  the  storage  of  electrical  energy 
in  the  form  of  chemical  energy.  Any  reversible  cell  may  be 
employed  as  an  accumulator.  Thus,  the  oxygen-hydrogen  cell 

Pto2  —  Solution  of  Sulphuric  Acid  —  Ptn2 

may  be  used  as  an  accumulator,  if  the  gases  resulting  from  the 
electrolysis  of  water  are  collected  at  the  electrodes  and  then  used 
to  produce  a  current.  Practically,  the  lead  accumulator  is  used 
almost  exclusively.  If  two  lead  plates  are  immersed  in  a  20  per 
cent  solution  of  sulphuric  acid,  a  minute  amount  of  lead  sulphate 
will  be  formed  on  the  surface  of  each  plate.  If  now  a  current  of 
electricity  is  passed  through  the  solution,  the  lead  sulphate  on  the 


ELECTROMOTIVE  FORCE  489 

cathode  will  undergo  reduction  to  metallic  lead,  and  the  lead  sul- 
phate on  tho  anode  will  be  oxidized  to  lead  peroxide.  In  this  way 
we  form  the  cell 

Pb  -  20  per  cent  Sol.  H2SO4  -  Pb02, 

the  electromotive  force  of  which  is  about  2  volts.  The  amounts 
of  lead  and  lead  peroxide  produced  in  this  way  are  so  small  that 
the  cell  can  only  furnish  a  very  small  amount  of  electrical  energy. 
In  order  to  increase  its  capacity,  the  electrodes  should  be  given 
as  large  an  amount  of  surface  as  possible.  This  may  be  brought 
about  by  the  method  of  Plante,  in  which  the  solution  is  elec- 
trolyzed  first  in  one  direction  and  then  in  the  other,  thus  causing 
the  plates  to  become  spongy;  or  by  the  method  of  Faure,  in  which 
a  lead  "grid  "  is  charged  with  a  paste  of  lead  oxide  and  red  lead, 
and  is  then  introduced  into  the  solution  and  the  current  passed 
until  we  obtain  spongy  lead  at  the  cathode  and  lead  peroxide  at  the 
anode.  If  now  the  charging  circuit  is  broken  and  the  two  elec- 
trodes are  connected  by  a  wire,  a  current  will  flow  from  the  peroxide 
plate  to  the  lead  plate,  lead  sulphate  being  slowly  formed  at  each. 
In  charging  the  accumulator,  the  lead  sulphate  on  the  negative 
electrode  is  reduced  to  metallic  lead,  the  reaction  being  represented 
by  the  following  equation :  — 

PbS04  +  2 (-)  =  Pb"  +  SO/'. 

At  the  positive  electrode  SO/'  ions  are  liberated  and  they  react 
with  the  lead  sulphate  and  the  water  of  the  electrolyte  in  the 
following  manner :  — 

PbS04  +  2  H20  +  SO/'  +  2  (+)  =  Pb02  +  4  H*  +  2  SO/'. 

When  the  cell  is  discharged  SO/'  ions  are  liberated  at  the  lead 
plate  forming  lead  sulphate,  as  shown  by  the  equation 

Pb  +  SO/'  +  2  (+)  =  PbS04. 

At  the  peroxide  plate  H*  ions  are  discharged  and,  in  the  presence 
of  the  electrolyte,  convert  the  lead  peroxide  into  lead  sulphate, 
according  to  the  equation 

PbO2  +  2  H*  +  H2SO4  +  2  (-)  =  PbSO4  +  2  H2O. 


490  THEORETICAL  CHEMISTRY 

Combining  the  foregoing  equations,  we  obtain  the  following  single 
equation  summarizing  the  chemical  changes  involved  in  the  pro- 
duction of  the  current:  - 

Pb02  +  Pb  +  2  H2S04  ^  2  PbS04  +  2  H20. 
The  upper  arrow  represents  the  reaction  on  discharging,  while 
the  lower  arrow  represents  the  reaction  on  charging. 

The  electromotive  force  of  the  storage  cell  is  approximately 
2  volts.  It  is  not  completely  reversible,  but  under  favorable 
conditions  its  efficiency  is  about  90  per  cent;  that  is,  90  per  cent 
of  the  electrical  energy  supplied  to  it  in  charging  can  be  recov- 
ered on  discharging.* 

PROBLEMS. 

1.  Calculate  the  heat  of  amalgamation  of  cadmium  at  0°  from  the 
following  data:  —  the  electromotive  force  of  a  cell  made  up  of  a  1-per  cent 
cadmium  amalgam  in  a  solution  of  cadmium  sulphate  is  0.06836  volt  at 
0°,  and  0.0735  volt  at  24°.45.  Ans.  510  calories  per  mol  of  Cd. 

2.  The  electromotive  force  of  the  cell 

Pb  -  0.01  m  Pb(N03)2  -  sat.  NH4N03  -  m  KC1,  HgCl  -  Hg 

is  -  0.469  volt  at  25°.  The  lead  nitrate  is  62  per  cent  ionized.  What 
is  the  potential  of  lead  against  a  solution  containing  1  mol  of  Pb  ions 
per  liter,  referred  to  the  calomel  electrodes  as  zero? 

Ans.  -  0.405  volt. 

3.  Calculate  the  electromotive  force  of  the  cell 

Cu  amalgam  (a)  —  Solution  CuS04  —  Cu  amalgam  (b), 

at  20°.8,  having  given  that  the  concentrations  of  the  amalgams  (a)  and 
(b)  are  0.0004472  and  0.00016645  respectively.  Ans.  0.0125  volt. 

4.  Calculate  the  electromotive  force  of  the  cell 

Cu  -  m  CuS04  -  0.01  m  CuS04  -  Cu, 

at  25°,  having  given  the  following  values  for  the  degree  of  ionization  of 
the  two  solutions:  —  for  m  copper  sulphate,  a  =  0.21,  and  for  0.01  m 

*  For  a  thorough  treatment  of  the  theory  of  the  lead  accumulator  the 
student  is  recommended  to  consult  "Die  Theorie  des  Bleiaccumulators  " 
by  F.  Dolezalek. 

For  a  detailed  account  of  the  primary  cells  in  common  use  the  reader  will 
find  Carhart's  "Primary  Batteries"  most  satisfactory. 


ELECTROMOTIVE  FORCE  491 

copper  sulphate,  a  =  0.61.  The  electromotive  force  at  the  junction  of 
the  two  solutions  may  be  neglected.  Am.  0.0458  volt. 

5.  At  25°  the  electromotive  force  of  the  cell 

Zn  -  0.5  m  ZnS04  -  0.05  m  ZnS04  -  Zn 

is  0.018  volt.  Neglecting  the  potential  developed  at  the  junction  of  the 
solutions,  and  assuming  the  dilute  solution  of  zinc  sulphate  to  be  ionized 
to  the  extent  of  35  per  cent,  find  the  degree  of  ionization  of  the  concen- 
trated solution.  Ans.  0.142. 

6.  What  is  the  electromotive  force  of  the  cell 

Zn  -  0.1  m  ZnS04  -  0.01  m  ZnS04  -  Zn, 

at  18°?    For  ZnS04,  -     -  =  0.601,  for  0.1  molar  ZnS04,  «  =  0.39,  and 

for  0.01  molar  ZnS04,  a  =  0.63.  Ans.  0.078  volt. 

7.  The  electromotive  force  of  the  cell 

Ag  -  0.001  m  AgN03  -  m  KN03  -  m  KI,  Agl  -  Ag 
is  0.22  volt  at  18°.    A  molar  solution  of  KI  is  78  per  cent  ionized,  and  a 
0.001  molar  solution  of  AgN03  is  98  per  cent  ionized.     Calculate  the  solu- 
bility of  Agl.  Ans.  3.53  X  10~3  mols  per  liter. 

8.  The  potential  of  zinc  against  a  solution  containing  one  mol  of 
Zn  ions  is  0.493  volt,  at  18°.    Assuming  complete  dissociation,  calculate 
the  solution  pressure  of  zinc  in  atmospheres. 

9.  The  electromotive  force  of  the  Daniell  cell 

Cu  -  CuS04  -  ZnS04  -  Zn 

is  1.0960  volt  at  0°,  and  1.0961  volt  at  3°.  Calculate  the  heat  of  the  reac- 
tion taking  place  in  the  cell.  Ans.  51,070  calories. 

10.  Calculate  the  electromotive  force  of  the  cell 

PtH2  -  0.1  m  KOH  -  m  HC1  -  PtHl) 

at  25°,  having  given  that  0.1  molar  KOH  is  85  per  cent  ionized  and  molar 
HC1  is  70  per  cent  ionized.  Ans.  0.757  volt. 


CHAPTER  XXL 
ELECTROLYSIS  AND  POLARIZATION. 

Polarization.  If  a  difference  of  potential  of  about  1  volt  is 
applied  to  two  platinum  electrodes  immersed  in  a  concentrated 
solution  of  hydrochloric  acid,  it  will  be  found  that  the  current 
which  passes  at  first,  steadily  diminishes  and  ultimately  becomes 
zero.  The  cessation  of  the  current  has  been  shown  to  be  due  to 
the  accumulation  of  hydrogen  on  the  cathode  and  chlorine  on  the 
anode,  these  two  gases  setting  up  an  opposing  electromotive  force 
called  the  electromotive  force  of  polarization. 

If  in  the  above  case  the  applied  electromotive  force  is  increased 
to  1.5  volts,  the  counter  electromotive  force  is  no  longer  sufficient 
to  reduce  the  current  to  zero.  In  fact,  at  any  voltage  above  1.35 
volts  a  continuous  current  passes;  this  is  termed  the  decomposition 
potential  of  hydrochloric  acid. 

At  all  voltages  above  the  decomposition  potential,  the  current  C 
may  be  calculated  by  means  of  the  formula 

E  -  e  =  CR, 

where  E  is  the  applied  electromotive  force,  e  the  counter  electro- 
motive force,  and  R  the  resistance  of  the  electrolyte. 

As  the  applied  electromotive  force  and  the  current  increase,  the 
polarization  increases,  since  the  gases  are  liberated  under  a  pres- 
sure greater  than  that  of  the  atmosphere;  but  since  the  gases 
escape  from  the  solution  the  value  of  e  can  never  become  equal 
to  E.  The  decomposition  potential  of  an  electrolyte  can  be 
determined  in  two  different  ways,  viz.:  (1)  by  gradually  raising 
the  applied  electromotive  force  E  until  it  exceeds  e,  when  the 
current  will  suddenly  increase;  or  (2)  by  charging  the  electrodes 
up  to  atmospheric  pressure  by  means  of  an  electromotive  force 
greater  than  e,  and  then  breaking  the  external  circuit  and  measur- 
ing the  counter  electromotive  force. 

492 


ELECTROLYSIS  AND  POLARIZATION 


493 


The  arrangement  of  apparatus  for  the  measurement  of  the 
electromotive  force  of  polarization,  as  suggested  by  Le  Blanc,* 
is  indicated  in  Fig.  106.  A  is  the  cell  in  which  polarization 
occurs,  B  is  the  source  of  external  electromotive  force,  C  is  a  capil- 
lary electrometer,  D  is  a  source  of  variable  potential,  and  E  is  one 
prong  of  an  electrically-driven  tuning  fork  which  serves  to  make 
and  break  contact  with  the  points  F  and  G  in  rapid  alternation. 
When  the  tuning  fork  makes  contact  at  F  the  polarizing  current 


B— 


Fig.  106. 


Fig.  107. 


flows  through  A,  polarizing  the  electrodes;  when  contact  is  made 
at  G,  the  counter  electromotive  force  due  to  this  polarization 
causes  a  current  to  flow  through  D  and  C.  The  counter  electro- 
motive force  is  balanced  by  varying  D,  until  C  indicates  zero 
current:  the  potential  of  D  is  then  equal  to  the  electromotive  force 
of  polarization.  Just  as  the  electromotive  force  of  a  galvanic  cell 
is  due  to  the  combined  action  of  several  differences  of  potential, 
so  also  the  electromotive  force  of  polarization  is  due  to  the  individ- 
ual differences  of  potential  located  at  the  electrodes.  The  method 
employed  for  the  measurement  of  polarization  at  a  single  elec- 
trode was  devised  by  Fuchs,  and  is  illustrated  diagrammatically 
in  Fig.  107.  Into  the  vessel  containing  the  electrolyte,  dip  the 
*  Zeit.  phys.  Chem.,  5,  469  (1890). 


494  THEORETICAL  CHEMISTRY 

two  electrodes  A  and  B,  and  the  side  tube  of  the  calomel  normal 
electrode  C.  D  is  the  source  of  external  electromotive  force,  E  is 
a  capillary  electrometer,  and  F  is  a  source  of  variable  potential. 
Before  closing  the  external  circuit  DAB,  the  potential  of  the 
electrode  B  against  the  solution  is  first  measured.  Then  the  cir- 
cuit DAB  is  closed,  thereby  polarizing  the  electrode  B.  The 
external  circuit  is  again  broken  and  the  potential  of  B  against  the 
solution  remeasured.  The  difference  between  the  final  and  initial 
values  of  the  electrode  potential  gives  the  polarization  at  B.  In 
like  manner,  the  polarization  at  the  electrode  A  can  be  measured. 
The  small  amount  of  electricity  which  is  necessary  to  polarize 
an  electrode  is  termed  the  polarization  capacity  of  the  electrode. 
This  factor  is  dependent  upon  the  extent  of  surface  of  the  electrode, 
and  also  upon  the  nature  of  the  metal  of  which  the  electrode  is 
made.  For  electrodes  of  equal  surface,  the  polarization  capacity 
of  palladium  is  greater  than  that  of  platinum,  when  hydrogen 
is  liberated  on  each.  The  solubility  of  hydrogen  is  greater  in 
palladium  than  in  platinum,  and  consequently,  because  a  larger 
amount  of  hydrogen  is  dissolved,  a  greater  quantity  of  electricity 
will  be  required  to  bring  the  pressure  of  the  hydrogen  up  to  that 
of  the  hydrogen  dissolved  in  the  platinum.  If,  through  the 
processes  of  solution  or  diffusion,  or  through  chemical  action,  the 
substance  which  causes  the  polarization  is  removed,  the  electrode 
is  said  to  be  depolarized.  Thus,  when  a  reducing  agent,  such  as 
ferrous  chloride,  is  electrolyzed,  the  oxygen  liberated  at  the  anode 
immediately  combines  with  the  electrolyte,  forming  ferric  chloride 
and  preventing  polarization  of  the  electrode. 

If  water  be  electrolyzed  between  platinum  electrodes,  the  cathode 
becomes  saturated  with  hydrogen  and  the  anode  with  oxygen, 
until,  when  tne  electromotive  force  of  polarization  becomes  equal 
to  that  of  the  external  circuit,  the  current  ceases.  The  two 
gases,  hydrogen  and  oxygen,  are  soluble,  however,  and  conse- 
quently diffuse  away  from  the  electrodes,  either  escaping  from 
the  solution  or  recombining  to  form  water.  In  order  to  compen- 
sate for  this  continuous  loss  of  gas  at  the  electrodes,  a  small  current 
continues  to  flow,  thus  maintaining  the  initial  electromotive  force 
constant.  This  small  current  is  termed  the  residual  current. 


ELECTROLYSIS  AND  POLARIZATION 


495 


If  oxygen  is  bubbled  over  the  surface  of  the  cathode  during  elec- 
trolysis, the  hydrogen  is  removed  as  rapidly  as  it  is  liberated. 
Such  an  electrode  on  which  no  new  substance  is  formed  during 
electrolysis  is  called  an  unpolarizable  electrode. 

Decomposition  Potentials.  The  decomposition  potential  of  an 
electrolyte  can  be  determined,  as  has  already  been  pointed  out, 
by  immersing  two  platinum  electrodes  in  the  solution  and  connect- 
ing with  a  source  of  electricity,  the  electromotive  force  of  which 
can  be  varied  at  will.  The  voltage  is  gradually  increased  and  the 
corresponding  current  is  observed.  The  current  increases  at  first 


Electromotive  Eonce 
Fig.  108. 

and  then  drops  almost  to  zero  every  time  the  voltage  is  raised, 
until  the  decomposition  potential  is  reached.  Beyond  this  point 
the  current  is  directly  proportional  to  the  electromotive  force. 
If  the  applied  electromotive  forces  are  plotted  as  abscissae  and 
the  corresponding  currents  as  ordinates,  we  obtain  curves  of  the 
form  shown  in  Fig.  108.  Some  of  the  decomposition  potentials 
of  molar  solutions  determined  by  Le  Blanc  *  are  given  in  the  accom- 
panying tables. 

*  Zeit.  phys.  Chem.,  8,  299  (1891). 


496 


THEORETICAL  CHEMISTRY 
SALTS. 


Salt. 

Decomp. 
Potential. 

Salt. 

Decomp. 
Potential. 

ZnSO4  

Volts. 
2.35 

Cd(NO3)2.. 

Volts. 
1.98 

ZnBr2 

1  80 

CdSO4 

2  03 

NiSO4 

2  09 

CdCl2 

1  88 

NiCl2 

1  85 

CoSO4 

1  92 

Pb(NO3)2 

1  52 

CoCl2 

1  78 

AgN03  

0.70 

ACIDS. 


Acid. 

Decomp. 
Potential. 

H2SO4. 

Volts. 
67 

HNO3.  . 

69 

H3PO4  

.70 

CH2C1.COOH  

.72 

CHC12.COOH  
CH2(COOH)2  
HC1O4  
HC1  
(COOH)2  
HBr  
HI  

.66 
.69 
.65 
.31 
0.95 
0.94 
0.52 

BASES. 


Base. 

Decomp. 
Potential. 

NaOH 

Volts. 
1   69 

KOH  
NH4OH  

1.67 

1.74 

It  will  be  observed  that  while  there  is  considerable  variation  in 
the  decomposition  potentials  of  salts,  there  is  very  little  variation 
in  the  decomposition  potentials  of  acids  and  bases.  There  is  a 
maximum  value  of  about  1.70  volts  to  which  many  acids  and 
bases  closely  approximate.  It  is  found  that  all  acids  and  bases 
which  decompose  at  1.70  volts  give  off  hydrogen  and  oxygen  at 


ELECTROLYSIS  AND  POLARIZATION 


497 


the  electrodes.  Those  acids  and  bases  which  decompose  at 
potentials  less  than  the  maximum,  do  not  liberate  hydrogen  and 
oxygen.  When  their  solutions  are  sufficiently  diluted,  however, 
hydrogen  and  oxygen  are  evolved  and  the  decomposition  potential 
rises  to  the  maximum  value.  Thus,  Le  Blanc  found  the  following 
values  for  the  decomposition  potential  of  different  dilutions  of 
hydrochloric  acid. 


Concentration. 

Decomp. 
Potential. 

2mHCl 

Volts. 

1.26 

imHCl 

1.34 

imHCl 

1.41 

AmHCl 

1.62 

AmHCl 

1.69 
/ 

When  2  m  hydrochloric  acid  is  electrolyzed,  hydrogen  and 
chlorine  are  given  off  at  the  electrodes,  whereas  when  the  concen- 
tration of  the  acid  is  reduced  to  1/32  m,  hydrogen  and  oxygen 
are  the  products  of  electrolysis,  and  the  decomposition  potential 
increases  to  1.70  volts.  It  is  found  that  the  values  of  the  decom- 
position potentials  vary  slightly  with  the  nature  of  the  electrodes 
used.  The  above  values  were  determined  with  platinum  elec- 
trodes. 

The  Theory  of  Polarization.  Our  knowledge  of  the  process 
taking  place  at  the  electrodes  during  electrolysis  is  largely  due 
to  the  investigations  of  Le  Blanc.  He  determined  the  electro- 
motive force  of  polarization  at  each  electrode,  varying  the  external 
electromotive  force  from  zero  up  to  the  decomposition  potential 
of  the  solution.  When  the  decomposition  value  was  reached,  he 
found  the  potential  of  the  electrode  against  the  solution  to  be  the 
same  as  the  difference  of  potential  between  the  solution  and  the 
element  liberated  at  the  electrode.  Thus,  the  decomposition 
potential  of  a  molar  solution  of  zinc  sulphate  is  2.35  volts;  the 
corresponding  difference  of  potential  between  the  electrode  and 
the  solution  is  found  to  be  0.493  volt.  If  a  piece  of  pure  zinc 
is  immersed  in  a  molar  solution  of  zinc  sulphate,  the  difference 


498  THEORETICAL  CHEMISTRY 

of  potential  is  found  to  be  0.493  volt,  the  metal  being  negative 
to  the  solution.  It  frequently  happens  that  the  electrode  exhibits 
the  potential  due  to  the  deposited  metal  before  the  decomposition 
point  of  the  solution  is  reached.  For  example,  in  a  molar  solution 
of  silver  nitrate  the  electrode  acquires  the  potential  of  pure  silver 
in  molar  silver  nitrate  before"  the  decomposition  value,  0.70  volt, 
is  reached.  This  is  due  to  the  negative  solution  pressure  of  the 
silver  which  causes  the  deposition  of  the  ions  of  the  metal  without 
the  application  of  any  external  electromotive  force.  When  an 
indifferent  electrode,  such  as  platinum,  is  immersed  in  a  solution 
of  a  salt,  a  very  small  amount  of  ionic  deposition  must  occur, 
otherwise,  according  to  the  Nernst  equation,  an  infinite  electro- 
motive force  must  be  established.  Thus,  in  the  equation 

RT        P 


if  the  solution^  pressure  P  =  0,  it  is  evident  that  TT  =  oo  and  a 
perpetual  motion  must  result.  We  are  thus  forced  to  the  con- 
clusion that  when  an  indifferent  electrode  is  immersed  in  a  salt 
solution,  ions  will  continue  to  separate  upon  it  until  the  tendency 
for  the  deposited  metal  to  go  back  into  solution  in  the  ionic  state 
exactly  counterbalances  the  tendency  to  separation.  Hence,  the 
electrode  will  become  positive  toward  the  solution.  The  magnitude 
of  this  difference  of  potential  will  be  dependent  upon  the  amount 
of  -metal  deposited.  It  is  to  be  noted  that  this  difference  of 
potential  need  not  be  equal  to  that  between  the  massive  metal 
and  the  solution.  If  the  electrodes  be  connected  with  an  external 
source  of  electromotive  force,  the  value  of  which  can  be  varied  at 
will,  and  a  small  electromotive  force  be  applied,  more  metal  will 
separate  on  the  cathode.  This  will  cause  an  increase  in  the  solu- 
tion pressure  P,  tending  to  offset  further  deposition.  A  still 
further  increase  in  the  external  .  electromotive  force  will  cause  the 
deposition  of  more  metal,  and  as  a  result  of  the  corresponding 
increase  in  P,  further  deposition  at  that  voltage  will  be  pre- 
vented. Ultimately,  when  the  applied  electromotive  force  is 
such  that  P  acquires  its  maximum  value,  equivalent  to  that  of 
the  massive  metal,  continuous  deposition  will  occur.  An  exactly 


ELECTROLYSIS  AND  POLARIZATION  499 

analogous  process  takes  place  at  the  anode.  If  a  gas  is  liberated, 
its  concentration  steadily  increases  until  the  maximum  pressure 
is  reached,  when  it  will  escape  from  the  solution.  When  strong 
currents  are  employed,  P  does  not  remain  constant,  as  has  been 
assumed  above,  but  gradually  diminishes  causing  the  difference  of 
potential  at  the  electrode  to  increase. 

From  the  above  considerations  it  becomes  clear  why  a  definite 
electromotive  force  is  necessary  to  bring  about  a  continuous 
decomposition  of  an  electrolyte:  this  will  only  take  place  when 
the  concentrations  of  the  substances  separating  at  the  electrodes 
have  attained  their  maximum  values.  When  the  decomposition 
point  is  reached,  the  electrode  exhibits  the  potential  characteristic 
of  the  massive  metal.  It  is  evident  from  the  behavior  of  silver 
nitrate  and  the  salts  of  other  metals  having  negative  solution 
pressures,  that  the  maximum  values  of  concentration  at  the 
the  electrodes  need  not  necessarily  be  attained  simultaneously. 

When  the  products  of  electrolysis  are  gaseous,  the  value  of  the 
decomposition  potential  depends  upon  the  nature  of  the  electrodes. 
Thus,  the  cell 

PtHj  -  m  H2S04  -  Pto2 

gives  an  electromotive  force  of  1.07  volts  if  platinized  platinum 
electrodes  are  used.  If  an  external  electromotive  force  slightly 
greater  than  1.07  volts  be  applied  to  this  cell  in  the  reverse  direc- 
tion, water  will  be  steadily  decomposed,  hydrogen  and  oxygen 
being  evolved  at  the  electrodes.  If  on  the  other  hand,  the  plati- 
nized electrodes  are  replaced  by  electrodes  of  polished  platinum, 
the  decomposition  potential  rises  to  1.68  volts.  The  reverse 
electromotive  force  of  polarization,  however,  is  only  1.07  volts. 
That  is,  the  liberation  of  gas  at  a  polished  platinum  electrode  is 
an  irreversible  process.  The  difference  in  the  behavior  of  the 
two  electrodes  is  explicable  when  it  is  remembered  that  platinum 
is  capable  of  occluding  large  amounts  of  gas.  A  platinized  elec- 
trode absorbs  the  liberated  gas  very  slowly,  and  when  thoroughly 
saturated,  if  it  is  not  entirely  immersed  in  the  solution,  it  gradually 
gives  up  the  gas  by  diffusion,  no  bubbles  being  formed.  Thus,  if 
the  external  electromotive  force  be  raised  to  1.07  volts,  the  system 


500  THEORETICAL  CHEMISTRY 

will  be  in  equilibrium,  while  if  the  applied  electromotive  force  be 
greater  or  less  than  the  equilibrium  value,  a  current  will  flow  in 
one  direction  or  the  other,  gas  being  either  liberated  or  dissolved. 
In  other  words,  the  cell  is  completely  reversible. 

Where  polished  platinum  or  gold  electrodes  are  used,  however, 
the  decomposition  potential  is,  as  has  been  stated,  1.68  volts. 
Polished  electrodes  have  relatively  small  absorbing  power.  Hence, 
if  an  electromotive  force  between  1.07  and  1.68  volts  be  applied, 
the  gases  cannot  diffuse  away  from  the  electrode  rapidly  enough, 
and,  when  the  solution  in  the  vicinity  of  the  electrodes  becomes 
saturated  with  gas,  the  current  ceases  to  flow. 

A  very  slow  process  of  diffusion  from  the  solution  into  the  air 
is  constantly  taking  place  however,  and  this  permits  the  continu- 
ous evolution  of  an  exceedingly  small  amount  of  gas,  while  a  corre- 
spondingly small  current  traverses  the  solution. 

In  order  to  produce  a  steady  electrolysis,  it  is  necessary  to  raise 
the  external  electromotive  force  to  such  an  intensity  that  it  is 
able  to  bring  about  the  formation  of  bubbles  at  the  surface  of  the 
electrodes.  This  calls  for  the  expenditure  of  an  amount  of  work 
depending  upon  the  condition  of  the  electrode  surfaces,  the  sur- 
face tension  of  the  solution,  and  various  other  factors.  In  cases 
where  bubbles  are  formed,  a  portion  of  the  available  energy  of 
the  chemical  process  is  not  expended  in  effecting  electrical  separa- 
tion; consequently  the  reverse  electromotive  force  is  less  than  the 
applied,  and  the  system  is  irreversible. 

The  reactions  at  the  electrodes  are  catalytically  accelerated  by 
the  metal  of  which  the  electrodes  are  made.  Thus,  platinized 
platinum  is  the  most  effective  catalyst  for  the  reaction  represented 
by  the  equation 

H2  +  2(+)<=±2H'. 

Hydrogen  is  liberated  on  platinized  platinum  at  the  potential 
0  volt,  on  polished  platinum  at  0.09  volt,  and  on  zinc  at  0.70  volt. 
The  electromotive  force  necessary  to  overcome  the  resistance  of 
the  chemical  reaction  at  an  electrode  is  termed  the  overvoltage. 
Thus,  we  say  that  hydrogen  is  liberated  on  polished  platinum  with 
an  overvoltage  of  0.09  volt,  and  on  zinc  with  an  overvoltage  of 


ELECTROLYSIS  AND   POLARIZATION 


501 


0.70  volt.  The  following  table  gives  the  overvoltage  necessary 
for  the  liberation  of  hydrogen  and  oxygen  on  electrodes  of  differ- 
ent metals. 

ELECTRODE  OVERVOLTAGES. 


Hydrogen  Liberation. 

Oxygen  Liberation. 

Metal. 

Overvoltage. 

Metal. 

Overvoltage. 

Pt  (platinized)  
Au  

0.00 
0.01 
0.08 
0.09 
0.15 
0.21 
0.23 
0.46 
0.53 
0.64 
0.70 
0.78 

Au  

.75 
.67 
.65 
.65 
.63 
.53 
.48 
.47 
.47 
.36 
.35 
.28 

Pt  (polished)  
Pd  
Cd 

Fe  (in  NaOH)  
Pt  (polished)  
Ag  
Ni  
Cu                  

Ag 

Pb. 

Cu. 

Pd  

Fe 

Sn  
Pb  
Zn  
Hg  

Pt  (platinized)  .  . 
Co... 

Ni  (polished)  
Ni  (spongy)  

Primary  Decomposition  of  Water  in  Electrolysis.  The  decom- 
position potential  of  an  electrolyte  giving  off  hydrogen  and  oxygen 
at  the  electrodes,  is  dependent  upon  the  concentrations  of  the  two 
ions,  H*  and  OH',  and  is  independent  of  the  nature  of  the  electrolyte. 
As  has  already  been  stated,  the  decomposition  potential  of  all 
acids  and  bases  giving  off  hydrogen  and  oxygen  approximates  to 
1.70  volts.  According  to  the  law  of  mass  action,  the  product  of 
the  concentrations  of  the  H*  and  OH'  ions  is  constant  and  inde- 
pendent of  the  other  substances  which  may  be  present;  hence, 
although  the  potentials  of  the  individual  electrodes  may  differ 
considerably,  their  sum  remains  practically  constant. 

Excluding  solutions  of  salts  which  undergo  reduction  by  hydro- 
gen, and  solutions  of  chlorides,  bromides,  and  iodides  reducible 
by  oxygen,  the  ions  H*  and  OH',  according  to  Le  Blanc,  are  to 
be  regarded  as  the  sole  factors  in  the  electrolysis  of  solutions,  and 
not  the  ions  of  the  dissolved  electrolyte.  In  other  words,  elec- 
trolysis involves  a  primary  decomposition  of  water. 

The  electrical  conductance  of  the  solution  is  due  to  the  ions  of 
the  electrolyte  together  with  the  ions  of  water,  but  at  the  electrode 


•502  THEORETICAL  CHEMISTRY 

that  process  takes  place  which  involves  the  expenditure  of  the 
minimum  amount  of  energy,  and  this  is,  under  ordinary  conditions 
the  separation  of  the  H"  and  OH'  ions. 

Thus,  when  a  solution  of  potassium  sulphate  is  electrolyzed, 
only  a  moderately  strong  current  being  used,  it  is  not  rational  to 
assume  the  discharge  of  the  K"  and  SO4"  ions  at  the  electrodes, 
and  then  subsequent  reaction  between  these  discharged  ions  and 
water.  This  may  be  made  clear  by  considering  the  process  taking 
place  at  the  cathode.  According  to  the  explanation  based  upon 
so-called  ''secondary  action,"  the  K"  ions  give  up  their  positive 
charges  to  the  electrode  and  then  react  with  water  as  indicated 
by  the  equation 

K  +  H-  +  OH'  -»K'  +  OH'  +  H. 

This  explanation  involves  the  transfer  to  the  potassium  atom  of 
the  positive  charge  of  the  H*  ion  of  water;  this  can  only  take  place 
if  the  H*  ion  holds  its  charge  less  tenaciously  than  the  K"  ion. 
Hence,  if  the  H"  ion  parts  with  its  charge  more  readily  than  the 
K'  ion,  the  former  will  be  discharged  primarily  at  the  cathode. 
Similar  reasoning  may  be  employed  to  explain  the  action  at  the 
anode.  Therefore,  in  electrolysis  all  of  the  ions  participate  in 
conducting  the  current  and  collect  around  the  electrodes,  but 
since  the  H*  and  OH'  ions  separate  more  easily,  these  are  dis- 
charged. With  stronger  currents  it  is  possible  to  cause  the  separ- 
ation of  the  K*  and  S04"  ions  also,  since  the  number  of  H*  and  OH' 
ions  present  is  too  small  to  carry  all  of  the  current,  and  the  energy 
required  to  discharge  the  ions  of  the  electrolyte  is  less  than  that 
necessary  to  remove  the  small  number  of  residual  H*  and  OH' 
ions.  The  formation  and  decomposition  of  water  are  reversible 
processes,  so  that  no  loss  of  energy  is  involved,  as  would  be  the 
case  if  secondary  actions  occurred. 

Electrolytic  Separation  of  the  Metals.  Freudenberg  *  was  the 
first  to  recognize  the  possibility  of  effecting  the  quantitative 
separation  of  different  metals  by  means  of  graded  electromotive 
forces.  He  showed  that  it  was  only  necessary  to  select  a  salt  of 
each  metal  the  decomposition  potentials  of  which  differ  as  widely 
as  possible,  and  electrolyze  at  an  electromotive  force  intermediate 
*  Zeit.  phys.  Chem.,  12,  97  (1893). 


ELECTROLYSIS  AND  POLARIZATION 


503 


between  these  potentials.  The  salt  having  the  lower  decomposi- 
tion potential  will  decompose  first,  and  when  the  deposition  of 
the  metal  is  complete,  the  current  will  practically  cease;  then  if 
the  applied  electromotive  force  be  raised  above  the  decomposition 
potential  of  the  second  salt,  the  second  metal  will  be  deposited. 
In  practice  it  is  foiind  necessary  to  increase  the  applied  electro- 
motive force  slightly  because  of  the  gradual  decrease  in  the  num- 
ber of  ions  of  the  salt  having  the  lower  decomposition  potential. 
The  amount  of  this  increase  may  be  readily  calculated  from  the 
familiar  equation 

RT .       P 

"K   —   — FT  lOffe  —  • 

nF      *  p 

Suppose  a  mixture  of  the  nitrates  of  cadmium,  lead  and  silver  is 
subjected  to  electrolysis,  the  decomposition  potentials  of  the  salts 
being  as  follows:  —  Cd(N03)2  =  1.98  volts,  Pb(N03)2  =  1.52 
volts,  and  AgNO3  =  0.70  volt.  The  applied  electromotive  force 
is  made  a  little  less  than  1  volt  and  all  of  the  silver  is  deposited ; 
then  the  electromotive  force  is  raised  to  about  1.6  volts,  thus 
depositing  all  of  the  lead;  and  finally,  with  an  electromotive  force 
of  about  2  volts  the  cadmium  is  deposited. 

In  the  subjoined  table  are  given  the  separation  potentials  of 
some  of  the  ions,  the  separation  potential  of  the  H*  ion  being 
assumed  to  be  equal  to  zero. 

SEPARATION  VALUES  OF  IONS  FOR  MOLAR  CONCENTRA- 
TION. 


Ion. 

Separation 
Potential. 

Ion. 

Separation 
Potential. 

Ae* 

-0  78 

I' 

0  52 

or" 

-0  34 

Br' 

0  94 

H* 

0  0 

0" 

1  08  (in  acid) 

Pb" 

0  17 

Cl' 

1  31 

Cd" 

0  38 

OH' 

1  68  (in  acid) 

Zn". 

0  74 

OH' 

0.88  (in  base) 

SO4" 

1.9 

According  to  this  table  the  decomposition  potential  of  water  is 
equal  to  the  sum  of  the  separation  potentials  of  its  ions,  or  1.68 
volts. 


CHAPTER  XXII. 
PHOTOCHEMISTRY. 

Radiant  Energy.  The  visible  portion  of  the  spectrum  is  com- 
prised between  the  extreme  red  at  one  end  and  the  extreme  violet 
at  the  other;  the  wave-length  corresponding  to  the  former  is 
approximately  0.7  micron,  while  that  corresponding  to  the  latter 
is  about  0.4  micron.  The  visible  portion  of  the  spectrum,  how- 
ever, is  but  a  small  fraction  of  the  entire  spectrum.  Beyond  the 
red  of  the  visible  spectrum  lies  the  region  of  the  so-called  infra- 
red, comprising  all  wave-lengths  from  0.76  micron  up  to  300  mi- 
crons. Beyond  the  infra-red,  between  300  and  2000  microns,  is 
an  unmeasured  region,  which  is  succeeded  by  the  region  of  elec- 
trical waves,  extending  from  2000  microns  to  an  undetermined 
maximum.  On  the  other  hand,  extending  beyond  the  violet  of 
the  visible  spectrum  is  the  so-called  ultra-violet  or  actinic  region, 
comprising  all  wave-lengths,  between  0.4  micron  and  0.1  micron. 
It  thus  appears  that  heat,  light,  and  electricity  are  all  forms  of 
radiant  energy,  the  only  distinction  between  them  being  a  differ- 
ence in  wave-length.  Very  little  is  known  concerning  radiant 
energy,  and  up  to  the  present  time  all  attempts  to  resolve  it  into 
a  capacity  and  an  intensity  factor  have  failed.  Whatever  may  be 
the  nature  of  this  form  of  energy,  we  know  that  the  effects  produced 
by  it  are  dependent  upon  the  wave-length  of  the  radiation. 

We  have  already  devoted  several  chapters  to  the  consideration 
of  thermochemistry  and  electrochemistry,  and  it  now  remains  to 
study  very  briefly  the  connection  between  chemical  energy  and 
that  subdivision  of  radiant  energy  called  light.  This  branch  of 
theoretical  chemistry  is  termed  photochemistry.  The  ultra-violet 
or  actinic  rays  are  the  most  active  chemically,  although  light  of 
every  wave-length,  including  the  invisible  infra-red,  is  capable  of 
producing  chemical  action.  When  light  falls  upon  a  substance,  a 
portion  of  the  incident  radiation  is  reflected,  a  portion  is  absorbed, 
and  a  portion  is  transmitted.  It  has  been  shown  that  only  that 

504 


PHOTOCHEMISTRY  505 

portion  of  the  incident  radiation  which  is  absorbed  is  effective  in 
producing  chemical  change. 

Radiant  energy  has  been  shown  by  Lebedew,*  and  also  by 
Nichols  and  Hull,t  to  exert  a  definite,  though  extremely  small 
pressure.  Thus,  the  pressure  of  solar  radiations  on  the  earth  is 
equivalent  to  that  of  a  column  of  mercury  1.4  X  10~9  mm.  high. 

Source  of  Radiant  Energy.  According  to  the  electromagnetic 
theory  of  light,  the  emission  of  waves  of  light  from  a  material 
source  is  due  to  the  vibrations  of  minute  charged  particles  called 
radiators.  These  radiators,  which  may  be  either  atoms  or  elec- 
trons, give  rise  to  electromagnetic  waves  of  the  same  period  as 
their  own,  that  is,  to  light  waves  of  definite  length.  The  energy 
required  to  produce  these  electromagnetic  waves  is  derived  from 
the  vibrating  system  itself,  and  unless  an  equivalent  amount  of 
energy  is  constantly  supplied  to  the  system,  the  amplitude  of  the 
vibrations  will  steadily  diminish  and  ultimately  cease. 

There  are  two  different  ways  in  which  this  supply  of  energy  can 
be  maintained.  First,  the  temperature  of  the  vibrating  system 
as  a  whole  may  be  kept  high.  This  type  of  radiation,  which  is 
maintained  by  purely  physical  means,  is  called  pure  temperature 
radiation.  Every  substance  whose  temperature  is  above  the  abso- 
lute zero  (  —  273°)  gives  rise  to  pure  temperature  radiation.  The 
higher  the  temperature,  the  more  rapid  and  the  more  energetic  the 
atomic  and  electronic  vibrations  become.  With  increase  in  rapid- 
ity of  vibration,  there  results  a  corresponding  diminution  in  wave- 
length, so  that,  as  the  temperature  is  raised,  the  longer  heat  waves 
are  succeeded  by  the  shorter  waves  of  the  visible  region  of  the 
spectrum.  When  a  sufficiently  high  temperature  is  reached,  the 
period  of  vibration  becomes  so  rapid  as  to  cause  the  radiation  of 
waves  corresponding  to  the  entire  range  of  the  visible  spectrum. 
At  higher  temperatures,  the  rate  of  vibration  is  such  that  the 
vibrating  particles  must  of  necessity  possess  extremely  small  mass: 
it  is  commonly  believed  that  under  these  conditions  the  vibration 
is  wholly  electronic. 

The  second  way  in  which  energy  may  be  supplied  to  the  vibrating 

*  Rapp.  pres  au  Congres  de  Physique,  2,  133  (1900). 
t  Phys.  Rev.,  13,  293  (1901). 


506  THEORETICAL  CHEMISTRY 

atoms  or  electrons  is  by  chemical  or  electrical  means.  The  general 
term  luminescence  has  been  proposed  by  Wiedemann  for  all  cases 
where  luminous  energy  is  derived  from  other  sources  than  high 
temperature.  It  is  to  be  observed  that  luminescence  is  frequently 
exhibited  by  systems  whose  temperatures  are  comparatively  low. 
For  example,  notwithstanding  the  fact  that  the  flame  resulting 
from  the  combustion  of  carbon  disulphide  has  a  temperature  of 
only  150°,  it  has  been  found  to  be  capable  of  affecting  the  photo- 
graphic plate.  Pure  temperature  radiation  alone  at  150°  would 
correspond  to  long  waves  in  the  infra-red  region  of  the  spectrum 
and,  as  is  well  known,  such  waves  are  incapable  of  exerting  appre- 
ciable photographic  action. 

Emission  and  Absorption.  The  relation  between  the  emissive 
and  absorptive  powers  of  different  bodies  was  first  clearly  enun- 
ciated by  Kirchhoff  *  in  1859.  This  law  may  be  stated  as  follows: 
—  Light  of  any  given  wave-length  emitted  by  a  body  can  also  be 
absorbed  by  the  same  body  at  a  lower  temperature.  This  law,  it  will 
be  seen,  offers  a  satisfactory  explanation  of  the  Fraunhofer  lines  in 
the  solar  spectrum.  The  sun  is  surrounded  by  a  gaseous  atmos- 
phere resulting  from  the  vaporization  of  the  elements  present  in  the 
body  of  the  sun.  Each  element  in  the  cooler  gaseous  envelope, 
according  to  KirchhofFs  law,  absorbs  those  wave-lengths  which  it 
emits  at  the  higher  temperature  of  the  solar  nucleus.  The  result- 
ing dark  lines  of  the  solar  spectrum  have  enabled  the  astronomer  to 
determine  the  elementary  composition  of  the  sun. 

If  the  emissive  and  absorptive  powers  of  a  body  be  denoted  by 
E  and  A  respectively,  then  according  to  Kirchhoff's  law, 

E/A  =  S, 

where  S  is  a  constant.  When  absorption  is  complete,  A  is  unity 
and  S  =  E.  Under  these  conditions  the  constant,  S,  may  be  de- 
fined as  the  emissivity  of  a  body  which  absorbs  all  of  the  incident 
radiation  and  reflects  none.  Such  a  body  was  called  by  Kirch- 
hoff a  perfectly  black  body.  The  emissivity  of  a  perfectly  black 
body  is  equal  to  the  ratio  of  the  emissive  to  the  absorptive  power  of 
any  body  at  the  same  temperature.  A  familiar  qualitative  illus- 

*  Ostwald's  Klassiker,  No,  100  (1898). 


PHOTOCHEMISTRY  507 

tration  of  KirchhofFs  law  is  that  afforded  by  the  appearance  of 
a  fragment  of  white  chinaware  possessing  a  dark  pattern  when 
heated  to  a  high  temperature.  The  dark  parts  of  the  design  absorb 
light,  while  the  white  parts  reflect  it.  On  heating  the  fragment  to 
redness,  the  pattern  will  be  reversed,  the  dark  portions  of  the 
design  appearing  bright  and  the  white  portions  dark. 

It  has  been  shown  that  the  law  of  Kirchhoff  is  a  necessary  con- 
sequence of  the  application  of  the  second  law  of  thermodynamics 
to  the  thermal  equilibrium  within  an  enclosure  whose  walls  are 
impervious  to  heat. 

The  Stefan-Boltzmann  Law.  From  a  study  of  the  experi- 
ments of  Dulong  and  Petit  on  the  rate  of  cooling  of  different  bodies, 
Stefan  *  discovered  an  empirical  relation  between  the  total  radia- 
tion of  a  body  and  its  temperature.  Later,  Boltzmann  f  derived 
the  same  relation  thermodynamically  and  showed  that  instead  of 
being  general,  as  Stefan  supposed,  it  is  only  strictly  applicable 
to  a  perfectly  black  body.  The  Stefan-Boltzmann  law  may  be 
stated  as  follows :  —  The  total  radiation  from  a  perfectly  black  body 
is  directly  proportional  to  the  fourth  power  of  the  absolute  tempera- 
ture. If  the  total  radiation  be  denoted  by  S,  we  may  write 

S  =  CT4, 

where  C  is  a  constant.  If  the  radiation  from  the  sun  be  con- 
sidered solely  as  a  temperature  effect,  its  temperature  may  be 
calculated  by  the  above  equation  expressing  the  Stefan-Boltzmann 
law.  Employing  available  bolometric  data,  the  temperature  of 
the  sun  may  thus  be  shown  to  be  6200°  absolute. 

The  Displacement  Law  of  Wien.  Having  considered  the  total 
energy  radiated  by  a  given  source,  we  now  come  to  the  considera- 
tion of  the  distribution  of  energy  throughout  the  entire  spectrum 
in  its  relation  to  temperature.  The  results  of  the  experiments 
of  Lummer  and  Pringsheim  J  on  the  distribution  of  energy  in  the 
normal  spectrum  of  a  black  body  are  shown  by  the  curves  of  Fig. 
109.  The  values  of  the  energy  radiated  by  the  source  are  plotted 

*  Sitz.  Ber.  Wiener  Akad.,  79  (II),  391  (1879). 
t  Wied.  Ann.,  22,  291  (1884). 

*  Verb,  deutsch.  phys.  Ges.,  i,  230  (1889);  3,  36  (1901). 


508 


THEORETICAL  CHEMISTRY 


as  ordinates  against  the  corresponding  values  of  the  wave-length 
as  abscissae.  It  will  be  observed  that  the  energy  corresponding  to 
a  definite  wave-length  increases  with  the  temperature,  and  that 
each  curve,  or  isothermal,  exhibits  a  distinct  maximum.  The 


1646° 


Wave-length  f 
Fig.  109. 

position  of  this  maximum  is  displaced  in  the  direction  of  decreasing 
wave-length  as  the  temperature  is  raised. 

In  1893,  Wien  *  discovered  the  law  governing  this  displacement 
of  the  energy  maximum  with  temperature.      If  Xmax.  denotes  the 
*  Wied.  Ann.,  58,  662  (1896). 


PHOTOCHEMISTRY  509 

wave-length  corresponding  to  the  energy  maximum,  and  T  is  the 
absolute  temperature  of  the  radiating  black  body,  Wien  showed 
that  \max  x  T  =  constant. 

Furthermore,  Wien  found  that 

Smax.  =  BT*, 

where  £max.  is  the  maximum  emissivity  corresponding  to  Xmax.. 
In  general,  the  emissivity  S  is  the  amount  of  energy  radiated  per 
second  from  a  narrow  strip  of  the  spectrum  corresponding  to  the 
mean  wave-length  X.  It  should  be  mentioned  that  while  the 
experimental  realization  of  a  perfectly  black  body  is  impossible, 
a  very  close  approximation  can  be  obtained  by  the  employment  of 
a  hollow  blackened  sphere  perforated  by  a  small  hole  to  permit  the 
passage  of  the  radiation.  When  this  sphere  is  heated,  practically 
all  of  the  radiation  is  absorbed  by  multiple  reflections  at  the  inner 
surface.  It  has  been  found  that  where  such  a  black  body  is  not 
available,  or  where  extreme  accuracy  is  not  required,  a  thin  strip 
of  platinum  foil  coated  with  ferric  oxide  and  heated  electrically 
proves  a  satisfactory  substitute.  It  is  hardly  necessary  to  call 
attention  to  the  fact  that  the  term  black  body,  as  here  used,  does 
not  imply  a  total  absence  of  color.  At  high  temperatures,  a 
"  black  "  source  of  radiation  may  be  red  or  even  white.  To  avoid 
confusion,  it  has  been  proposed  to  substitute  the  term  full  radia- 
tor for  the  older  term,  black  body. 

Distribution  of  Energy  throughout  the  Spectrum.  The  experi- 
mental determination  of  the  radiant  energy  corresponding  to  a 
given  wave-length,  really  resolves  itself  into  the  measurement  of 
the  energy  emitted  between  two  contiguous  wave-lengths,  X  and 
X  +  d\.  In  other  words,  we  actually  measure  the  energy  of  a 
very  small  portion  of  the  spectrum  included  between  two  wave- 
lengths which  lie  very  close  together. 

Several  different  formulae  have  been  proposed  for  the  calcula- 
tion of  the  distribution  of  energy  throughout  the  spectrum.  Of 
these,  the  formulae  of  Rayleigh  and  Wien  have  been  found  to 
reproduce  experimental  values  with  considerable  accuracy.  The 
formula  of  Rayleigh  has  the  following  form: 


510  THEORETICAL  CHEMISTRY 

while  that  of  Wien  may  be  written  thus: 


In  these  two  formulas,  C  and  c'  are  constants,  while  the  other 
symbols  have  their  usual  significance.  The  formula  of  Rayleigh 
has  been  found  to  hold  better  in  the  region  of  the  longer  wave- 
lengths while  the  reverse  is  true  of  the  formula  of  Wien.  It  should 
be  remembered  that  both  of  these  formulas  apply  only  to  bodies 
emitting  continuous  spectra. 

High  Temperature  Thermometry.*  Various  optical  methods 
for  the  measurement  of  high  temperatures  have  been  developed, 
but  a  detailed  treatment  of  these  methods  is  obviously  out  of  place 
in  a  book  of  this  character.  Mention  should  be  made,  however, 
of  two  instruments  which  have  proven  of  great  value  in  high  tem- 
perature measurements. 

The  optical  pyrometer  of  Fe*ry  is  based  upon  Stefan's  law  of 
total  radiation.  It  consists  of  a  telescope  fitted  with  an  objective 
of  fluorite,  at  the  focus  of  which  is  placed  a  sensitive  thermo- 
couple. In  order  to  determine  the  temperature  of  a  source  of 
radiant  energy,  such  as  a  crucible  of  molten  metal,  the  telescope 
is  directed  toward  the  contents  of  the  crucible  and  the  image  is 
focussed  on  the  thermo-j  unction  by  means  of  an  adjustable  eye- 
piece. The  resulting  electric  current  is  then  measured  by  means 
of  a  galvanometer. 

In  the  optical  pyrometer  of  Holborn  and  Kurlbaum,  use  is  made 
of  the  luminous  radiations  only.  In  this  instrument  the  current 
through  a  small  incandescent  lamp  is  varied  until  its  light  is  just 
eclipsed  by  that  from  the  hot  body.  When  this  point  of  balance 
has  been  reached,  the  incandescent  filament  and  the  hot  body 
have  the  same  temperature.  The  pyrometer  is  calibrated  by  de- 
termining the  current  necessary  to  raise  the  filament  of  the  lamp 
to  the  temperature  of  the  standard  black  body,  the  temperature 
of  the  latter  being  determined  by  means  of  a  thermocouple.  Of 

*  For  a  detailed  account  of  optical  pyrometers,  the  student  is  referred  to 
"High  Temperature  Measurements"  by  Le  Chatelier  and  Boudouard,  trans- 
lated by  Burgess  (John  Wiley  and  Sons,  Inc.). 


PHOTOCHEMISTRY  511 

course,  when  the  instrument  is  used  to  determine  the  temperature 
of  sources  of  radiant  energy  which  differ  widely  in  character  from 
that  of  a  perfectly  black  body,  the  accuracy  of  the  measure- 
ments is  lessened,  but  even  in  an  extreme  case,  such  as  that  pre- 
sented by  polished  platinum  at  950°,  the  error  does  not  exceed 
74°.  The  importance  of  optical  pyrometers  in  photochemical 
investigations  lies  chiefly  in  the  determination  of  energy  curves 
of  light  sources  and  in  the  absolute  measurement  of  radiant 
energy. 

Luminescence.  As  has  already  been  pointed  out,  it  is  cus- 
tomary to  distinguish  between  pure  temperature  radiation  and 
luminescence.  The  latter  term  is  applied  to  all  cases  where  chemi- 
cal or  electrical  energy  is  transformed  directly  into  radiant  energy. 
The  various  types  of  luminescence  may  be  conveniently  classified 
in  the  following  manner:  — 

Type  of  Luminescence.  Origin  of  Radiation. 

(1)  Photo-luminescence  Preliminary  exposure  of  the  lumi- 

(a)  Fluorescence,  nous    substance    to    some    external 

(6)  Phosphorescence.  source  of  radiant  energy. 

(2)  Thermo-luminescence.  Stimulation  by  heat,  but  at  a  tem- 

perature considerably  lower  than  that 
required  for  pure  temperature  radia- 
tion. 

(3)  Chemi-Iuminescence.  Chemical  reaction. 

(4)  Tribo-luminescence.  Fracture  or  cleavage  of  crystals. 

(5)  Cathodo-luminescence  Electric  discharge. 

(6)  Radio-luminescence.  Radioactivity. 

By  the  term  fluorescence  is  meant  the  phenomenon  of  the  emission, 
by  an  illuminated  medium,  of  light  of  a  different  wave-length  from 
that  of  the  incident  radiation.  In  general,  the  wave-length  of  the 
transformed  radiation  is  greater  than  that  of  the  incident  radiation. 
This  law,  to  which  several  exceptions  have  been  discovered,  was 
first  enunciated  by  Stokes.  When  the  incident  radiation  is  cut 
off,  fluorescence  ceases. 

On  the  other  hand,  there  are  many  substances  which  continue 
to  emit  light  for  some  time  after  the  external  light-stimulus  is 
removed.  This  phenomenon  is  termed  phosphorescence  and 
appears  to  be  governed  by  Stokes  law  for  fluorescence.  The 


512  THEORETICAL  CHEMISTRY 

property  of  phosphorescence  appears  to  be  limited  to  anhydrous 
substances. 

Among  the  numerous  substances  which  are  known  to  exhibit  the 
phenomenon  of  fluorescence  may  be  mentioned  fluorite  (from 
which  the  phenomenon  derived  its  name),  uranium  glass,  petro- 
leum, solutions  of  organic  dyestuffs,  and  quinine  sulphate.  The 
vapors  of  sodium,  mercury,  and  iodine  have  recently  been  found 
by  Wood  to  fluoresce  brilliantly. 

The  sulphides  of  the  alkaline  earths  may  be  mentioned  as  ex- 
amples of  phosphorescent  substances.  The  investigations  of 
Lenard  and  Urbain  have  revealed  the  interesting  fact  that  the 
presence  of  a  trace  of  one  of  the  heavy  metals  greatly  intensifies 
the  light  emitted  by  a  phosphorescent  substance. 

The  phenomenon  of  thermo-luminescence  calls  for  little  com- 
ment. There  seems  to  be  an  intimate  connection  between  thermo- 
luminescence  and  phosphorescence,  since  the  substances  exhibit- 
ing the  former  phenomenon  must  be  exposed  initially  to  light, 
otherwise  they  do  not  emit  any  visible  radiation  on  gentle  heating. 

Chemi-luminescence  is  a  phenomenon  accompanying  many 
chemical  reactions.  Thus,  the  precipitation  of  sodium  chloride 
from  its  saturated  solution  by  hydrochloric  acid  gas  is  accompanied 
by  an  emission  of  light  which  may  readily  be  seen  if  the  reaction  is 
carried  out  in  a  dark  room. 

When  certain  crystals,  such  as  those  of  cane  sugar,  are  either 
crushed  or  simply  rubbed  together,  flashes  of  light  are  emitted. 
This  phenomenon  is  known  as  tribo-luminescence. 

Cathodo-  and  radio-luminescence  may  be  considered  as  sub- 
divisions of  the  more  inclusive  term,  electro-luminescence.  At- 
tention has  already  been  called  to  the  fact  that  the  residual  gas  in 
a  vacuum  tube  is  rendered  luminous  by  the  passage  of  the  electric 
discharge,  and  also  that  certain  minerals  become  phosphorescent 
when  placed  in  the  path  of  the  cathode  rays.  These  may  be  taken 
as  examples  of  cathodo-luminescence.  The  luminosity  of  a  screen 
coated  with  crystals  of  zinc  sulphide,  when  subjected  to  the  action 
of  the  a-particles  shot  out  from  a  radio-active  substance,  has  also 
been  mentioned  in  a  previous  chapter.  This  is  clearly  an  instance 
of  radio-luminescence. 


PHOTOCHEMISTRY  513 

Having  briefly  reviewed  the  different  processes  involved  in  the 
production  of  light  we  now  turn  to  a  consideration  of  the  chemical 
phenomena  resulting  from  exposure  to  light. 

Photochemical  Action.  The  development  of  the  green  color 
of  plants  under  the  influence  of  the  rays  of  the  sun,  and  the  reverse 
process  of  bleaching  in  darkness,  were  probably  the  first  photo- 
chemical reactions  to  be  observed.  To-day  it  is  known  that  light 
has  the  power  of  initiating  or  accelerating  every  variety  of  chemical 
change.  This  statement  may  be  illustrated  by  the  following  typi- 
cal photochemical  reactions :  —  The  polymerization  of  anthracene, 
the  depolymerization  of  ozone,  the  transformation  of  maleinoid 
into  fumaroid  forms,  the  hydrolysis  of  acetone,  the  oxidation  of  lead 
sulphide,  and  the  reduction  of  silver  salts.  That  such  a  variety  of 
photochemical  reactions  should  result  from  exposure  to  mixed,  or 
heterogeneous,  light  is  due  to  the  selective  absorption  of  each  par- 
ticular chemical  system. 

Photochemical  action  is  not  limited  to  the  waves  of  the  visible 
spectrum  alone,  but  extends  from  the  red  end  of  the  spectrum 
(wave-length  800  MM)  *  into  the  ultra-violet  region  (wave-length 
300  MM)-  In  fact,  the  shorter  wave-lengths  of  the  ultra-violet 
region  of  the  spectrum  have  been  found  to  be  the  most  active 
photochemically . 

Laws  of  Grotthuss.  The  two  fundamental  generalizations 
of  photochemistry  were  first  enunciated  by  Grotthuss  in  1818. 
These  generalizations  may  be  stated  as  follows :  — 

(1)  Only  those  rays  of  light  which  are  absorbed  produce  chemical 
action. 

(2)  The  action  of  a  ray  of  light  is  analogous  to  that  of  a  voltaic  cell. 
It  has  recently  been  shown  by  Bancroft  |  that  the  second  of 

these  two  laws  is  inadequate  to  account  for  all  of  the  known  facts 
and,  therefore,  he  proposes  the  following  modification: — All  of 
the  radiations  which  are  absorbed  by  a  substance  tend  to  eliminate 
that  substance.  It  is  merely  a  question  of  chemistry  whether  any  re- 
action occurs  and  what  the  products  of  the  reaction  will  be. 

*  1  MM~7  =  10~7  cm. 

t  Jour.  Phys.  Chem.,  12,  209,  318,  417  (1908);  13,  1,  181,  269,  449,  538 
(1909);  14,  292  (1910). 


514  THEORETICAL  CHEMISTRY 

Quantitative  Relations  Concerning  the  Absorption  of  Light. 

When  a  ray  of  light  enters  an  absorbing  medium  only  a  certain 
proportion  of  the  incident  radiation  is  absorbed.  The  intensity  of 
the  light  entering  an  absorbing  medium  is  not  equal  to  that  which 
is  incident  on  the  surface  of  the  medium,  owing  to  the  fact  that  a 
portion  of  the  incident  beam  is  reflected.  It  has  been  found  that 
if  the  thickness  of  the  medium  be  increased  in  arithmetical  pro- 
gression, the  intensity  of  the  transmitted  light  decreases  in  geomet- 
rical progression.  If  the  intensity  of  the  light  traversing  a  layer 
dl  be  denoted  by  7,  then 

-  dl/dl  =  kl,  (1) 

where  k  is  a  constant  depending  upon  the  nature  of  the  absorbing 
medium  and  the  wave-length  of  the  light.  This  constant  k  is 
known  as  the  absorption  index.  If  the  initial  intensity  of  the  light 
is  /o  and  the  total  thickness  of  the  medium  is  d,  equation  (1)  be- 
comes, on  integrating, 

/  =  /0  .  e-kd.  (2) 

Or  equation  (2)  may  be  written  in  the  form, 


If  we  replace  e~k  by  a,  then  equation  (2)  becomes 

///o  =  of.  (4) 

The  constant  a  is  called  the  transparency  or  the  transmission  co- 
efficient. Bunsen  and  Roscoe  introduced  the  term  extinction  co- 
efficient. This  quantity  may  be  defined  as  the  reciprocal  of  that 
thickness  of  the  medium  which  reduces  the  intensity  of  the  trans- 
mitted light  to  one-tenth  of  its  initial  value.  If  the  extinction 
coefficient  be  denoted  by  e,  its  value  may  be  calculated  from  equa- 
tion (2)  in  the  following  manner  :  — 

r=  i«  .  io-«*,  (5) 

or  6=3logeJT  (6) 

It  is  evident  that  erk  is  identical  with  10~£  or  e  =  mk,  where  m 
represents  the  modulus  of  the  Naperian  system  of  logarithms. 


PHOTOCHEMISTRY  515 

In  1852  Beer  *  enunciated  an  important  law  concerning  the  in- 
fluence of  concentration  on  absorption.  Beer's  law  may  be  stated 
as  follows  :  —  The  absorption  of  light  by  different  concentrations  of 
ike  same  solute  dissolved  in  the  same  solvent  is  an  exponential  function 
of  the  concentration,  provided  the  thickness  of  the  absorbing  medium 
be  maintained  constant.  It  follows  from  this  law  that 


(7) 

where  c  is  the  concentration  of  the  solution. 

If  Beer's  law  is  valid,  the  ratio,  c/e  =  A,  known  as  the  absorption 
ratio,  should  be  constant.  Beer's  law  has  been  tested  with  a  large 
number  of  solutions  and  has  been  found  to  hold  quite  generally 
where  no  change  in  the  solute  occurs  when  the  concentration  of  the 
solution  is  altered. 

Photochemical  Extinction.  According  to  the  first  law  of 
Grotthuss,  only  the  absorbed  light  is  chemically  active.  The  con- 
verse of  this  law,  viz.,  that  every  substance  which  absorbs  light 
undergoes  chemical  change,  apparently  does  not  hold.  Further- 
more, only  a  portion  of  the  rays  absorbed  by  a  light-sensitive  sub- 
stance are  directly  involved  in  effecting  chemical  change.  For 
example,  while  an  alkaline  copper  tartrate  solution  shows  marked 
absorption  in  the  infra-red,  red,  yellow,  and  ultra-violet,  it  has 
been  proven  that  the  photochemical  reduction  of  the  copper  salt 
to  cuprous  oxide  is  due  to  the  action  of  the  ultra-violet  rays  alone. 

The  first  quantitative  measurements  of  the  absorption  of  light 
by  a  reacting  system  were  made  by  Bunsen  and  Roscoe.f  These 
investigators  found  that  the  absorption  of  light  by  hydrogen  and 
chlorine,  taken  separately,  was  less  than  that  of  the  reacting 
mixture  of  the  two  gases.  From  this  result  they  concluded  that 
in  a  photochemical  reaction,  the  absorption  of  light  by  the  re- 
acting system  is  greater  than  the  sum  of  the  individual  absorptions 
of  the  reacting  substances.  The  absorption  of  light  by  the  re- 
acting system,  over  and  above  the  ordinary  or  "optical  "  absorp- 
tion of  the  reacting  substances,  they  called  photochemical  extinc- 
tion. While  the  phenomenon  of  photochemical  extinction  is  re- 

*  Pogg.  Ann.,  86,  78  (1852). 
t  Ostwald's  Klassiker,  No.  38. 


516 


THEORETICAL  CHEMISTRY 


garded  by  many  physical  chemists  as  having  doubtful  significance, 
nevertheless  the  distinction  between  purely  optical  absorption 
on  the  one  hand,  and  chemical  absorption  on  the  other,  is  quite 
generally  accepted.* 

If  the  distinction  drawn  by  Bunsen  and  Roscoe  between  optical 
and  chemical  absorption  be  accepted,  then  all  photochemical  re- 
actions may  be  regarded  as  belonging  to  one  or  the  other  of  two 
classes,  as  follows:  —  (1)  reactions  in  which  light  does  work 
against  chemical  affinity,  the  work  being  equivalent  to  the  photo- 
chemical extinction;  or  (2)  reactions  in  which  the  light  functions 
merely  as  a  catalyst.  In  reactions  belonging  to  the  first  class,  the 
light  is  considered  to  be  the  agent  which  actually  initiates  chemical 
change,  whereas  in  reactions  of  the  second  class,  the  light  is 
assumed  to  accelerate  reactions  which  would  otherwise  proceed  at 
a  slower  rate. 

Kinetics  of  Photochemical  Reactions.  Let  A  and  B  repre- 
sent two  chemically  distinct  substances,  and  let  us  assume  that  the 
reaction 


takes  place  under  the  influence  of  light.     The  course  of  the  re- 
action can  be  followed  in  the  usual  manner  by  determining  the 

amount  of  either  constituent 
which  is  present  at  any  definite 
time.  There  are  certain  factors, 
however,  which  render  such  kin- 
etic measurements  more  difficult 
in  the  case  of  a  photochemical 
reaction  than  in  that  of  an  ordi- 
nary chemical  reaction.  Let  us 
assume  that  the  above  reaction 


•Fig.  110. 


is  homogeneous  and  of  the  first  order  and  also  that  it  takes  place 
in  homogeneous  solution.  It  is  apparent  that  if  the  system  is 
illuminated  from  one  side,  as  shown  in  Fig.  110,  the  rate  at  which 
A  undergoes  transformation  into  B  will  depend  upon  the  thick- 

*  For  a  condensed  summary  of  the  different  theories  of  light-absorption, 
the  student  should  consult  Sheppard's  "  Photochemistry  "  (Longmans,  Green 
&  Co.). 


PHOTOCHEMISTRY  517 

ness,  d,  of  the  absorbing  layer.  Hence,  if  the  velocity  of  the 
transformation  be  denoted  by  dB/dt,  we  may  write 

dB/dt  =  k[A], 

in  which  the  value  of  the  constant  k  will  not  only  be  a  function  of 
the  intensity  and  wave-length  of  the  light  but  also  of  position. 
Furthermore,  the  variation  in  the  velocity  of  the  reaction  in  suc- 
cessive layers  will  cause  differences  in  concentration  which  will  tend 
to  become  equalized  by  the  process  of  diffusion.  From  this  ex- 
ample it  is  obvious  that  the  extent  to  which  light  is  absorbed  is 
dependent  upon  the  dimensions  of  the  absorbing  system. 

If  light  be  regarded  as  a  material  substance,  then  we  may  con- 
veniently consider  its  absorption  as  analogous  to  the  diffusion  of  a 
gas  into  a  liquid  and  the  light  intensity  at  any  point  as  the  analogue 
of  concentration  or  active  mass.  According  to  the  law  of  mass 
action,  the  velocity  of  a  chemical  reaction  is  expressed  by  the 
equation, 

dx/dt  =  /ccini  >ef*' .-.  .  --  k'cin*'  •  c^'  .  .  . 

where  Ci,  Cz,  etc.,  are  the  concentrations  of  the  substances  entering 
into  the  reaction,  and  n\,  n2,  etc.,  are  the  coefficients  derived  from 
the  chemical  equation  and  indicating  the  order  of  the  reaction.  In 
applying  this  equation  to  photochemical  reactions,  the  following 
possibilities  must  be  borne  in  mind :  — 

(1)  The  reaction  may  take  place  in  successive  stages.     Under 
these  conditions  the  experimentally  measured  velocity  will  corre- 
spond to  that  of  the  slowest  reaction. 

(2)  Side  reactions  may  occur  with  the  formation  of  products 
quite  different  from  those  resulting  from  the  main  reaction. 

(3)  There  may  be  catalysis.     In  fact,  this  phenomenon  is  fre- 
quently met  with  in  the  study  of  photochemical  reactions. 

Nernst  has  pointed  out  that  the  velocity  constants,  k  and  k',  in 
the  foregoing  equation,  may  be  conveniently  considered  as  being 
directly  proportional  to  the  intensity  of  the  light,  for  light  of  the 
same  kind.  Owing  to  absorption,  this  intensity  will  be  a  function 
of  position  in  the  absorbing  medium. 

In  applying  the  law  of  mass  action  to  photochemical  reactions, 
it  is  important  to  note  that,  as  a  general  rule,  the  exponents  n\,  nt, 


518  THEORETICAL  CHEMISTRY 

etc.,  in  the  equation  of  the  so-called  dark-reaction  are  not  identical 
with  the  exponents  v\,  i>2,  etc.,  of  the  light-reaction.  In  other 
words,  the  order  of  a  chemical  reaction  is  usually  different  in  the 
light  from  what  it  is  in  the  dark.  The  photochemical  exponents 
v\,  vz,  etc.,  are  never  greater  than,  and  are  seldom  equal  to,  the 
corresponding  exponents  n\,  n?,  etc.,  of  the  dark-reaction. 

Thus,  Bodenstein  *  showed  that  the  dissociation  of  hydriodic 
acid,  in  the  dark,  is  a  reaction  of  the  second  order  and  may  be  rep- 
resented by  the  equation 

2  HI  <=>  H2  +  I2, 

whereas  when  hydriodic  acid  dissociates  in  the  light,  the  reaction 
is  of  the  first  order,  as  shown  by  the  equation 

HI  <±  H  +  I. 

In  this  case,  the  light  acts  as  a  catalyst,  merely  accelerating  the 
velocity  of  the  dark-reaction. 

The  action  of  light  on  the  reverse  reaction  may  be  such  as  to 
oppose  its  usual  course  in  the  dark.  In  this  case  the  resulting 
photochemical  equilibrium,  or  so-called  photo-stationary  state,  will 
not  be  identical  with  the  corresponding  chemical  equilibrium.  It 
should  be  observed  that  the  photo-stationary  state  differs  from 
ordinary  chemical  equilibrium,  in  that  its  permanency  is  wholly 
dependent  upon  the  constancy  of  the  source  of  illumination;  i.e., 
when  the  light  is  cut  off,  the  photo-stationary  state  shifts  to  the 
ordinary  chemical  equilibrium,  provided  the  reaction  is  reversible. . 
It  has  also  been  found  that  the  temperature  coefficients  of  most 
photochemical  reactions  are  negligible. 

Becquerel  f  discovered  that  silver  chloride,  which  had  been 
precipitated  in  the  dark,  was  only  sensitive  to  short  wave-lengths 
of  light,  whereas  silver  chloride,  which  had  been  exposed  for  a  few 
moments  to  sunlight,  became  sensitive  to  all  wave-lengths  rin  the 
visible  spectrum  and  to  the  shorter  wave-lengths  of  the  infra-red. 
This  phenomenon,  which  has  been  observed  with  various  sub- 
stances, was  first  studied  systematically  by  Bunsen  and  Roscoe  | 
who  termed  it  photochemical  induction. 

*  Zeit.  phys.  Chem.,  22,  23  (1897). 
t  Ann.  Chim.  Phys.  [3],  9,  257  (1843). 
j  Pogg.  Ann.,  100,  481  (1857). 


PHOTOCHEMISTRY  519 

Employing  a  mixture  of  hydrogen  and  chlorine  gases  they  found 
that  under  constant  illumination,  the  velocity  of  formation  of 
hydrochloric  acid,  which  was  hardly  appreciable  at  first,  increased 
rapidly  to  a  maximum  and  then  remained  constant.  The  interval 
of  time  required  for  the  reaction  to  attain  its  maximum  velocity 
is  known  as  the  period  of  induction. 

It  has  been  found  that  in  almost  every  photochemical  reaction 
there  is  a  similar  period  of  initial  perturbation.  The  phenomenon 
has  been  thoroughly  investigated  by  Burgess  and  Chapman  *  who 
arrived  at  the  conclusion  that  induction  effects  are  to  be  ascribed 
to  the  presence  of  minute  traces  of  various  impurities,  such  as 
gases  or  water  vapor,  adsorbed  by  the  walls  of  the  reaction-vessel. 
We  may  therefore  conclude  that  induction  effects  are  not  char- 
acteristic of  photochemical  reactions. 

Classification  of  Photochemical  Reactions.  According  to 
Sheppard  ("Photochemistry")  photochemical  reactions  may  be 
conveniently  classified  in  the  following  manner :  — 

(1)  Reversible  reactions,  i.e.,  reactions  in  which  the  products 
formed  under  the  influence  of  light  react  to  reproduce  the  original 
system  when  the  light  is  removed. 

(2)  Irreversible  reactions,  i.e.,  reactions  in  which  the  light  pro- 
motes transformation  to  a  more  stable  system.     Irreversible  re- 
actions are  subdivided  into  — 

(a)  Complete  reactions;  and 

(b)  Pseudo-reversible  reactions. 

The  polymerization  of  anthracene  may  be  taken  as  an  example 
of  a  reversible  photochemical  reaction.  This  reaction  may  be 
represented  as  taking  place  according  to  the  equation 

light 

2  Ci4Hi0  <=±  C2sH20, 

dark 

the  upper  arrow  indicating  the  direction  of  the  light-reaction  (poly- 
merization) and  the  lower  arrow  that  of  the  dark-reaction  (de- 
polymerization)  . 

I    An  illustration  of  a  completely  irreversible  photochemical  re- 
action is  afforded  by  a  mixture  of  hydrogen  and  chlorine  gases 
*  Jour.  Chem.  Soc.,  89,  1402  (1906). 


520  THEORETICAL  CHEMISTRY 

which  combine,  on  exposure  to  light,  to  form  hydrochloric  acid  gas, 
according  to  the  equation 

light 

H2  +  C12  -»  2  HC1. 

If  the  hydrochloric  acid  is  removed  by  solution  in  water  as 
fast  as  it  is  formed,  the  velocity  of  the  reaction  will  be  directly 
proportional  to  the  intensity  of  the  light.  Bunsen  and  Roscoe  * 
made  use  of  the  foregoing  facts  in  the  construction  of  their 
actinometer. 

The  reduction  of  ferric  oxalate  may  be  taken  as  an  example  of 
a  pseudo-reversible  photochemical  reaction.  This  substance  is 
reduced  to  ferrous  oxalate  on  exposure  to  light  as  shown  by  the 
equation 

light 

Fe2(C204)3  -»  2  Fe  (C2O4)  +  CO2. 

In  the  dark,  ferrous  oxalate  is  re-oxidized  to  ferric  oxalate  by  the 
oxygen  of  the  air.  While  the  initial  substance  is  reproduced,  it  is 
evident  that  the  reaction  is  not  strictly  reversible. 

Actinometers.  A  number  of  different  forms  of  apparatus  have 
been  devised  for  measuring  the  chemical  action  of  light:  such  in- 
struments are  known  as  actinometer s. 

The  hydrogen-chlorine  actinometer  is  based  upon  the  well- 
known  fact  that  the  speed  of  the  reaction  between  hydrogen  and 
chlorine  varies  greatly  with  the  intensity  of  illumination.  Bun- 
sen  and  Roscoe, f  guided  by  the  experiments  of  Draper,  con- 
structed an  actinometer  in  which  the  rate  of  combination  of  hydro- 
gen and  chlorine  could  be  measured  by  allowing  the  hydrochloric 
acid  formed  to  dissolve  in  water,  and  noting  the  resulting  diminu- 
tion of  volume.  A  diagram  of  this  apparatus  is  given  in  Fig.  111. 
The  apparatus  is  filled  with  a  mixture  of  equal  parts  of  hydrogen 
and  chlorine,  obtained  by  the  electrolysis  of  a  solution  of  hydro- 
chloric acid.  The  bulb  A,  containing  water,  is  connected  at  one 
end  with  a  tube  fitted  with  a  stop-cock  B,  and  at  the  other  end  with 
a  horizontal  tube  terminating  in  a  reservoir  D,  which  also  contains 
water.  When  the  water  has  become  saturated  with  the  constitu- 

*  "  Photochemische  Untersuchungen,"  Ostwald's  Klassiker,  No.  34. 
t  Fogg.  Ann.,  100,  43  (1857);  101,  235  (1857). 


PHOTOCHEMISTRY  521 

ents  of  the  gaseous  mixture,  B  is  closed  and  the  entire  apparatus  is 
protected  from  light.  When  it  is  desired  to  measure  the  photo- 
chemical action  of  a  source  of  light,  the  bulb  A  is  uncovered  and 
the  light  is  allowed  to  fall  upon  it.  Some  of  the  hydrogen  and 
chlorine  will  combine,  and  the  hydrochloric  acid  formed  will  be 
absorbed  by  the  water  in  A ;  the  column  of  water  in  the  horizontal 


Fig.  111. 

tube  will  move  to  the  right,  the  magnitude  of  the  movement  being 
measured  on  the  scale  C.  In  this  way  the  amount  of  photochemi- 
cal action  can  be  determined.  An  objection  to  the  use  of  hydrogen 
and  chlorine  in  the  actinometer  is  the  danger  of  violent  explosions 
when  the  illumination  is  too  intense.  To  remove  this  objection, 
Burnett  replaced  the  hydrogen  of  the  mixture  by  carbon  monoxide. 
Action  of  Light  on  the  Silver  Halides.  Probably  the  most 
familiar,  and  undoubtedly  one  of  the  most  important,  photo- 
chemical reactions  is  that  which  takes  place  on  the  exposure  of  a 
photographic  plate.*  Luther  f  has  shown  that  when  a  pure  silver 
halide  is  exposed  to  light,  it  undergoes  reduction  according  to  the 
equation 

light 

2  AgX  ->  Ag2X  +  X, 

where  X  may  be  chlorine,  bromine,  or  iodine.  On  removing  the 
light,  the  sub-halide  recombines  with  the  free  halogen  as  shown 
by  the  equation 

dark 

Ag2X  +  X  ->  2  AgX. 

In  other  words,  the  reaction  is  strictly  reversible  and  a  well- 
defined  photo-stationary  state  results  from  a  given  intensity  of 

*  For  an  excellent  treatment  of  the  chemistry  of  photography  the  student 
is  recommended  to  consult  "Photography  for  Students  of  Physics  and  Chem- 
istry," by  Louis  Derr  (Macmillan);  or  "Photochemie  und  Beschreibung  der 
photographischen  Chemikalien,"  by  H.  W.  Vogel. 

t  Zeit.  phys.  Chem.,  30,  628  (1899). 


522  THEORETICAL  CHEMISTRY 

illumination.  The  investigations  of  Baker  *  make  it  appear  quite 
probable  that  the  photochemical  reduction  of  the  silver  halides  is 
dependent  upon  the  presence  of  a  minute  trace  of  water  vapor  as  a 
catalyst. 

In  the  photographic  plate,  the  silver  halide  is  embedded  in  gela- 
tine, which  not  only  accelerates  the  rate  of  reduction  of  the  silver 
halide  but  also  causes  the  reaction  to  become  irreversible.  The 
alteration  in  the  behavior  of  the  silver  halide,  brought  about  by 
the  presence  of  gelatine,  is  due  to  the  fact  that  the  latter  reacts  with 
the  free  halogen  according  to  the  equation 

light 

2  AgBr  +  gelatine  — »  Ag2Br  +  brominated  gelatine. 

The  continuous  removal  of  the  liberated  bromine  by  the  gelatine 
is  an  example  of  what  is  known  as  photochemical  sensitization. 
Another  example  of  photochemical  sensitization  is  afforded  by  a 
mixture  of  benzene  and  silver  chloride.  The  normal  darkening 
of  the  silver  salt  is  markedly  increased  by  the  presence  of  the 
benzene,  which  combines  with  the  chlorine  as  rapidly  as  it  is  set 
free  by  the  action  of  the  light  on  the  silver  halide. 

While  the  silver  bromide  of  the  photographic  plate  is  extremely 
sensitive  to  the  shorter  wave-lengths  corresponding  to  the  violet  and 
ultra-violet  regions  of  the  spectrum,  it  is  only  reduced  by  the  longer 
wave-lengths  after  prolonged  exposure.  It  has  been  found,  how- 
ever, that  the  addition  of  certain  dyestuffs,  such  as  eosine  and 
Congo-red,  renders  the  silver  halide  sensitive  to  the  longer  wave- 
lengths of  the  spectrum.  This  phenomenon,  which  is  known  as 
optical  sensitization,  must  be  carefully  distinguished  from  chemical 
sensitization  to  which  reference  has  already  been  made.  It  is  to 
be  noted  that  an  optical  sensitizer  does  not  absorb  chemically  any 
of  the  products  of  the  reaction.  Photographic  plates  which  have 
been  sensitized  in  this  way  are  commonly  known  as  ortho-chro- 
matic plates. 

All  of  the  dyestuffs  which  can  function  as  optical  sensitizers 
have  been  found  to  exhibit  anomalous  refraction;  i.e.,  for  wave- 
lengths slightly  longer  than  those  absorbed,  these  substances 
possess  an  abnormally  large  refractive  index,  in  consequence  of 

*  Jour.  Chem.  Soc.,  61,  782  (1892). 


PHOTOCHEMISTRY  523 

which  the  refracted  waves  exert  the  same  effect  upon  the  silver 
halide  as  the  shorter  wave-lengths  of  the  spectrum.  Although  it 
is  not  an  essential  property,  it  is  generally  found  that  optical  sen- 
si  tizers  are  fluorescent. 

As  an  example  of  optical  sensitization,  where  the  sensitizer  is 
non-fluorescent,  we  may  take  the  photochemical  reduction  of 
mercuric  chloride  in  the  presence  of  ammonium  oxalate.  This  re- 
action takes  place  according  to  the  equation 

light 

2  HgCl2  +  (NH4)2C204  -»  Hg2Cl2  +  NH4C1  +  2  CO2. 

The  presence  of  the  non-fluorescent  ferric  ion,  Fe"",  has  been 
shown  by  Winther  *  to  be  an  effective  optical  sensitizer  in  the  re- 
action. The  ferric  ion  is  reduced  to  the  ferrous  state  while  the 
oxalic  acid  undergoes  oxidation  by  the  mercuric  chloride.  It 
was  pointed  out  by  Eder  f  that  this  reaction  is  well  adapted  for 
actinometric  measurements,  since  the  amount  of  mercurous  chlo- 
ride precipitated  is  directly  proportional  to  the  intensity  of  the 
light. 

Photochemical  After-Effect.  Certain  photochemical  reac- 
tions have  been  discovered  in  which  the  reaction  proceeds  even 
after  the  light  stimulus  is  removed.  This  phenomenon  is  known 
as  the  photochemical  after-effect.  The  velocity  of  a  reaction  of 
this  kind  is  different  from  that  of  either  the  light-  or  the  dark- 
reaction.  It  has  also  been  found  that  if  a  portion  of  the  reac- 
tion-mixture, which  has  already  been  exposed  to  the  light,  be 
added  to  a  fresh  unexposed  portion,  the  latter  immediately  com- 
mences to  decompose.  Thus,  a  solution  of  iodoform  in  chloro- 
form becomes  brown,  on  exposure  to  light,  due  to  liberation  of 
iodine. 

If  the  solution  is  removed  from  the  light,  the  decomposition  will 
continue  for  several  days,  and  if  a  small  portion  of  the  partially  de- 
composed solution  be  added  to  a  freshly  prepared  solution,  the 
latter  will  commence  to  decompose.  The  photochemical  after- 
effect can  be  readily  explained  if  we  assume  that  the  action  of  light 
gives  rise  to  heterogeneous  nuclei  which  persist  for  a  sufficient  time 

*  Zeit.  wiss.  Phot.,  7,  409  (1909). 
t  Sitz.  her.  Wien.  Akad.  (1879). 


524  THEORETICAL  CHEMISTRY 

after  the  light  is  removed  to  act  as  centers  around  which  the 
reaction  can  proceed  throughout  the  unexposed  portion  of  the 
mixture. 

Assimilation  of  Carbon  Dioxide.  A  photochemical  reaction 
of  special  interest  to  the  biologist  is  that  in  which  atmospheric 
carbon  dioxide  is  taken  up  by  plants  under  the  influence  of  solar 
radiation.  While  the  reaction  is  exceedingly  complex,  it  may  be 
regarded  as  taking  place  according  to  the  hypothetical  equation 

light 

6  C02  +  5  H2O  ->  C6H1005  +  6  02. 

(starch) 

This  reaction  represents  a  gain  in  energy  amounting  to  approxi- 
mately 685  calories  per  formula- weight  of  starch.  It  is  quite 
probable,  however,  that  the  initial  product  of  the  reaction  is  for- 
maldehyde and  that  subsequently  the  latter  substance  undergoes 
polymerization  with  the  formation  of  starch  and  other  carbo- 
hydrates. The  green  coloring  matter  of  the  leaves  of  plants, 
known  as  chlorophyll,  also  plays  an  important  part  in  the  assimi- 
lation of  carbon  dioxide,  but  beyond  the  fact  that  its  action  is  not 
catalytic,  little  can  be  stated  as  to  the  manner  in  which  it  functions 
in  the  reaction.  The  velocity  of  the  reaction  has  been  found  to 
attain  its  maximum  value  in  yellow  and  green  light,  a  result 
which  is  in  complete  agreement  with  the  fundamental  law  of  Grott- 
huss  that  only  those  rays  which  are  absorbed  are  active  chemi- 
cally. As  has  already  been  stated,  the  temperature  coefficients 
of  photochemical  reactions  are  generally  very  small.  This  is 
not  true,  however,  of  the  reaction  under  consideration.  It  has 
been  found  that  the  rate  at  which  carbon  dioxide  is  assimilated 
by  a  plant  is  nearly  doubled  for  a  rise  in  temperature  of  10° 
(see  p.  377). 

Photochemical  Synthesis.  While  it  has  long  been  known 
that  light  is  capable  of  effecting  the  synthesis  of  complex  organic 
compounds  in  the  living  plant  from  carbon  dioxide  and  water,  it 
is  only  recently  that  its  efficiency  in  bringing  about  a  great  variety 
of  organic  reactions  has  been  fully  recognized.  Owing  to  the  re- 
searches of  Ciamician  and  others  in  this  field,  numerous  photo- 
chemical syntheses  of  considerable  practical  value  to  the  organic 


PHOTOCHEMISTRY  525 

chemist  have  been  discovered.  It  must  suffice  here  to  mention  a 
few  typical  photochemical  syntheses.  Alcohols  may  be  oxidized 
in  successive  steps,  the  action  of  the  light  closely  resembling  the 
action  of  ferments.  Methyl  alcohol  and  acetone  react  to  form  iso- 
butylene  glycol  according  to  the  equation 

CH3     ^ht      CH3 
CH3  -  OH  +  CO  COH  -  CH2OH 

CH3  CH3 

The  action  of  light  has  been  found  to  be  especially  favorable  to  such 
processes  as  the  foregoing,  involving  reciprocal  oxidation  and  re- 
duction. 

Ortho-nitro-benzaldehyde  in  the  presence  of  ethyl  alcohol  reacts 
to  form  the  ethyl  ester  of  o-nitroso-benzoic  acid  as  shown  by  the 
equation 

N02      ,0  «*t      NO        ,0 

'\H          ^ 

+     CH2-OH~ 


Photoelectric  Cells.  The  absorption  of  light  by  any  one  of 
the  three  states  of  matter  is  invariably  accompanied  by  a  change 
in  electrical  condition.  The  different  electric  effects  accompany- 
ing the  absorption  of  radiant  energy  have  been  classified  by  Shep- 
pard  in  the  following  manner:  - 

(1)  lonization  with  a  corresponding  increase  of  conductance  in 
gases,  liquids,  and  solids  caused  by  transmission  of  light. 

(2)  Indirect  ionization  of  a  gas  due  to  reflection  or  emission  of 
electrons  from  the  surface  of  a  contiguous  denser  phase. 

(3)  Development  of  electromotive  forces  in  cells  of  the  following 
type:  — 


Electrode  A 

(light) 


Conductor, 
Di-electric 


Electrode  A 

(dark) 


where  the  two  electrodes  are  separated  by  a  medium  which  is  par- 
tially a  conductor  and  partially  a  di-electric. 


526  THEORETICAL  CHEMISTRY 

The  treatment  of  the  first  and  second  of  these  three  classes  of 
photoelectric  phenomena  properly  lies  within  the  domain  of  pure 
physics.  The  third  class,  however,  includes  a  number  of  photo- 
galvanic  combinations  of  considerable  interest  and  importance  to 
the  physical  chemist. 

The  first  investigation  of  photoelectric  combinations  was  under- 
taken by  E.  Becquerel  *  in  1839.  He  prepared  a  cell  consisting  of 
identical  plates  of  pure  silver,  coated  with  a  silver  halide,  and 
immersed  in  dilute  sulphuric  acid  as  an  electrolyte.  On  exposing 
one  electrode  to  the  light,  while  the  other  was  kept  in  the  dark,  an 
appreciable  electromotive  force  was  developed,  the  current  flowing 
in  the  solution  from  the  darkened  to  the  illuminated  electrode.  If 
a  galvanometer  be  included  in  the  circuit,  the  deflection  of  the 
needle  may  be  taken  as  a  measure  of  the  intensity  of  the  light. 

The  action  of  the  Becquerel  actinometer  has  been  explained  by 
Ostwald  as  follows :  —  The  incident  light  lessens  the  stability  of 
the  silver  iodide,  which  undergoes  ionization  according  to  the 
equation 

AgI->Ag+I': 

the  Ag*  ions  give  up  their  charges  to  the  electrode  while  the  I' 
ions  enter  the  solution.  For  every  Ag'  ion  which  is  discharged  at 
the  illuminated  electrode,  an  equal  number  of  Ag*  ions  enter  the 
solution  at  the  darkened  electrode,  thus  charging  the  latter  nega- 
tively. Hence  the  current  flows  in  the  solution  from  the  dark- 
ened to  the  illuminated  electrode.  Rigollot  f  has  constructed  an 
electrical  actinometer  in  which  the  silver  plates  used  by  Becque- 
rel are  replaced  by  two  oxidized  copper  electrodes,  while  instead 
of  dilute  sulphuric  acid  a  dilute  solution  of  sodium  chloride  is  used. 
A  number  of  photoelectric  combinations  have  been  investigated 
by  Wildermann,  J  among  which  he  recommends  the  following  as 
being  especially  suitable  for  actinometers :  — 

*  Ann.  Chim.  Phys.  [3],  9,  257  (1843). 

t  Jour,  de  Phys.  [3],  6,  520  (1897). 

j  For  a  thorough  treatment  of  theoretical  photochemistry,  the  student  is 
referred  to  Sheppard's  "Photochemistry,"  while  Plotnikow's  " Photochemische 
Versuchstechnik  "  is  recommended  as  an  excellent  guide  to  practical  laboratory 
methods. 


PHOTOCHEMISTRY  527 

(a)  Ag  |  AgBr,  0.1  N  KBr  |  AgBr  |  Ag 

(light)  (dark) 

Reaction,     2  AgBr  — >  Ag2Br, 

(b)  Cu  |  CuO  |  NaOH  |  CuO  |  Cu 

(light)  (dark) 

Reaction,     2  CuO  ->  Cu2O. 

Of  these  two  cells,  the  latter  is  perhaps  the  better,  since  it  gives  an 
appreciable  electromotive  force  for  light  of  the  same  intensity  and 
varying  wave-length. 

While  all  of  the  theoretical  interpretations  of  the  action  of  photo- 
electric cells  are  more  or  less  inadequate,  the  investigations  of 
Scholl  and  others  make  it  appear  probable  that  negative  electrons 
enter  the  electrolyte  from  the  electrode,  while  positive  ions  are 
discharged  on  the  electrode,  thereby  causing  anodic  polarization. 
In  general  the  direction  of  the  photoelectric  current  inside  the  cell 
is  from  the  non-exposed  to  the  exposed  electrode. 


INDEX  OF  NAMES 


Abegg,  218. 

Alexieeff,  172,  174. 

Amagat,  74,  75. 

Andrews,  105. 

Angstrom,  141. 

Arrhenius,  211,  227,  368,  377,  392, 

413,  425,  428,  432. 
Aston,  63,  419. 
Avogadro,  9,  76. 

Babo,  208. 

Baker,  521. 

Baly,  140,  141,  142. 

Barlow,  159. 

Barnes,  262. 

Bassett,  395. 

Bates,  430. 

Beccaria,  385. 

Bechhold,  259. 

Beckmann,  220. 

Becquerel,  42,  518,  526. 

Beer,  515. 

Bergmann,  312. 

Berkeley,  Earl  of,  197,  200,  201. 

Bernoulli,  76. 

Berthelot,  289,  308,  313,  323,  326. 

Berthollet,  4,  312. 

Berzelius,  16,  386. 

Bigelow,  381. 

Biltz,  260,  429. 

Bingham,  119. 

Biot,  132. 

Blagden,  217. 

Blake,  253,  254. 

Bodenstein,  317,  518. 

Boettger,  415. 


Boguski,  377. 

Boltwood,  44,  56. 

Boltzmann,  76,  507. 

Boyle,  2,  49,  72,  76,  170,  192,  197, 

286. 

Bragg,  W.  H.,  160. 
Bragg,  W.  L.,  160. 
Bredig,  276,  381,  409. 
Brown,  279, 
Bruhl,  127. 
Bruni,  250. 
Budde,  81, 
Buff,  121. 

Bunsen,  169,  170,  514,  515,  518,  520. 
Burgess,  519. 
Burton,  253. 

Cailletet,  113. 
Callendar,  201. 
Carlisle,  385. 
Chapman,  519. 
Charpy,  353. 
Ciamician,  524. 
Claude,  115. 
Clausius,  76,  392,  413. 
Clement,  96,  382,  383. 
Cotton,  252. 
Crookes,  33,  47,  48. 
Gumming,  472. 
Curie,  43,  44,  49,  56. 

Dale,  126,  152. 
Dalton,  1,  7,  16,  168. 
Davy,  385,  386. 
Debierne,  44. 
Debray,  328. 


529 


530 


INDEX  OF  NAMES 


Debye,  161. 

De  Chancourtois,  21. 

De  Forcrand,  334. 

Denham,  480. 

Desch,  142. 

Desormes,  96,  382,  383. 

Deville,  88,  321. 

De  Vries,  201,  202,  203,  204,  230. 

Dewar,  114. 

Dobereiner,  21. 

Draper,  520. 

Drude,  151. 

Duclaux,  248. 

Dulong,  11,  162. 

Dumas,  20. 

Dutoit,  419. 

Eder,  523. 
Einstein,  283,  285. 
Eotvos,  146. 

Fanjung,  418. 

Faraday,  7,  138,  150,  268,  389. 

Faure,  499. 

F&y,  510. 

Fick,  206,  208. 

Forbes,  466. 

Fletcher,  285. 

Freudenberg,  502. 

Freundlich,  254,  264,  269,  270. 

Friderich,  419. 

Fuchs,  493. 

Gay-Lussac,  8,  76,  192,  196,  197,  208. 

Geiger,  40,  52,  54,  55,  57. 

Geoffrey,  312. 

Gibbs,  273,  274,  338,  448. 

Gladstone,  126,  152. 

Goodwin,  468. 

Gouy,  279. 

Graham,  206,  237,  274,  276. 

Grotthuss,  391,  513,  524. 

Grove,  392. 

Guldberg,  314,  329. 

Guthrie,  349. 


Hall,  66. 

Hamburger,  204,  205. 
Hampson,  114. 
Hardy,  253,  256. 
Harkins,  63,  66. 
Hartley,  140,  142,  144,  197. 
Hautefeuille,  317. 
Hatty,  159. 
Hedin,  205. 
Helmholtz,  448,  477. 
Henry,  170,  184. 
Henderson,  472. 
Hess,  290,  304,  305. 
Heycock,  353. 
Heydweiller,  414. 
Hittorf,  393. 
Holborn,  478,  510. 
Horstmann,  331. 
Hulett,  181. 
Hull,  505. 

Isambert,  330. 

Jones,  219,  233,  395,  418. 
Jurin,  144. 

Kirchhoff,  506. 

Knoblauch,  375. 

Koelichen,  426. 

Kohlrausch,  400,  413,  414,  415,  416, 

417,  478,  488. 
Konowalow,  175,  176. 
Kopp,  14,  88,  120,  121,  127. 
Kraus,  430. 
Kroenig,  76. 
Kundt,  98. 
Kurlbaum,  510. 

Laborde,  56. 
Landolt,  4. 
Langbein,  302. 
Laplace,  98,  290. 
Larmor,  373. 
Lavoisier,  2,  286,  290. 
Lebedew,  505. 


INDEX  OF  NAMES 


531 


Le  Bel,  134,  135,  137. 

Le  Blanc,  493,  495,  497,  501. 

Le  Chatelier,  95,  309. 

Lehmann,  165. 

Lemoine,  317. 

Lenard,  36. 

Lewis,  162,  472,  474. 

Liebig,  382. 

Lillie,  249. 

Linde,  114. 

Lindemann,  164. 

Linder,  243,  251,  254. 

Lippmann,  459. 

Lodge,  411. 

Loomis,  218. 

Lorentz,  126,  152. 

Lorenz,  126,  152. 

Lessen,  121. 

Ludeking,  267. 

Lummer,  507. 

Lund&i,  442. 

Luther,  521. 

Mac  Innes,  472. 

Marignac,  5,  16. 

Maxwell,  76. 

Mayer,  94,  290. 

Mendeleeff,  22. 

Menschutkin,  378. 

Meyer,  466. 

Meyer,  Lothar,  22,  28. 

Meyer,  Victor,  83. 

Millikan,  40,  285. 

Mitscherlich,  14. 

Morgan,  149,  150. 

Morley,  16. 

Morse,  194,  196,  197,  198,  200. 

Moseley,  60. 

Mouton,  252. 

Natanson,  91. 

Nernst,  119,  141,  151,  164,  297,  335, 

336,  419,  451,  472,  475,  476,  486, 

517. 
Neumann,  13. 


Neville,  353. 
Newlands,  22. 
Newton,  312. 
Nicholson,  385. 
Nollet,  Abb6,  188. 
Nordlund,  285. 
Noyes,  372,  377. 
Nuttall,  52. 

Ohm,  388. 

Olszewski,  113. 

Onnes,  116. 

Ostwald,  W.,  8,  147,  169,  191,  231, 

273,  287,  368,  379,  383,  393,  422, 

424,  435,  447,  526. 
Ostwald,  Wo.,  245. 

Palmaer,  453. 

Pappada,  250. 

Pasteur,  132,  133,  134. 

Pauli,  254. 

Pebal,  90. 

Perkin,  138. 

Perrin,  35,  250,  279,  280,  281,  282, 

283,  285. 
Petit,  11,  162. 
Pfeffer,  188,  189,  190,  191,  192,  193, 

194,  198. 
Philip,  170. 
Pictet,  113. 
Picton,  243,  251,  254. 
Planck,  40,  207,  472,  474. 
Plant6,  499. 
Pope,  159. 
Pringsheim,  507. 
Proust,  5. 
Prout,  20. 
Pulfrich,  124. 

Quincke,  250. 

Ramsay,  49,  146,  147,  148,  149,  150, 

168. 
Raoult,  209,  210,  212,  215,  217,  218, 

223. 


532 


INDEX  OF   NAMES 


Rayleigh,  509. 
Regener,  40,  54. 
Regnault,  83. 
Reicher,  368. 
Reinitzer,  165. 
Reuss,  250. 

Richards,  16,  159,  466. 
Rigollot,  526. 

Roberts- Austen,  185,  35$,  355. 
Rodewald,  267. 
Roentgen,  42. 
Roozeboom,  351,  353. 
Roscoe,  177,  514,  515,  518,  520. 
Rose,  313. 
Royds,  55. 
Rudolphi,  430. 

Rutherford,  40,  45,  46,  48,  49,  50,  54, 
55,  57. 

Sargent,  472,  474. 

Schiff,  121. 

Scholl,  527. 

Schroeder,  265. 

Shields,  146,  147,  148,  149,  150. 

Siedentopf,  241. 

Snell,  124. 

Soddy,  48,  49,  50,  51,  52,  60. 

Soret,  208. 

St.  Gilles,  Pean  de,  313,  323. 

Stas,  16,  20. 

Stefan,  507. 

Steno,  155. 

Stokes,  39,  283. 

Svedberg,  277. 

Tammann,  206. 
Thilorier,  113. 
Thomsen,  302. 

Thomson,  J.  J.,  35,  45,  68,  282,  387, 
419. 


Thorpe,  121. 
Traube,  122,  129,  188. 
Trouton,  118. 

Van  der  Stadt,  373. 

Van  der  Waals,  31,  81,  82,  107,  121, 
171. 

Van't  Hoff,  135,  137,  184,  185,  192, 
193,  196,  197,  213,  217,  218,  223, 
226,  324,  337,  378,  412,  430. 

Volta,  385,  446. 

Waage,  314,  329. 
Walden,  404. 
Walker,  117,  211,  376. 
Warder,  368. 
Washburn,  417. 
Weber,  12,  13. 
Weimarn,  245. 
Wenzel,  312. 
Whetham,  412. 
Whitney,  253,  254,  377. 
Wiedemann,  250,  267. 
Wien,  508,  509. 
Wiener,  279. 
Wildermann,  526. 
Williamson,  392. 
Wilson,  C.  T.  R.,  45. 
Wilson,  E.  D.,  63. 
Winther,  523. 
Wladimiroff,  205. 
Wood,  512. 
Wroblewski,  113. 
Wullner,  208. 

Young,  183. 
Zsigmondy,  241,  259. 


INDEX  OF  SUBJECTS 


Abnormal  solutes,  225. 
Absolute  index  of  refraction,  125. 
Absorption  and  emission,  506. 
Absorption  index,  514. 
Absorption  of  light,  quantitative  re- 
lation concerning,  514. 
Accumulators,  488. 
Actinometers,  520. 
Active  deposits,  49. 
Activity  and  concentration,  478. 
Adsorption,  267. 

of  gases,  268. 

in  solution,  269. 

isotherm,  269,  270. 
Alloys,  353. 
Amicrons,  239. 
Association  in  solution,  225. 
Atomic  theory,  6,  7. 

number,  58. 

structure,  57. 

structure,  hydrogen-helium  system 
of,  63. 

weights,  16,  18. 

weights  and  atomic  numbers,  re- 
lation between,  64. 
Autocatalysis,  381. 
Avogadro  constant,  40. 

hypothesis  of,  9,  79. 

Basicity  of  organic  acids,  434. 
Bimolecular  reactions,  366. 
Boiling  point  constant,  213. 

and  critical  temperature,  119. 

elevation  of,  213. 

elevation    and    osmotic    pressure, 
216. 


Blood  corpuscle  method,  204. 
Brownian  movement,  279. 
recent  investigations  of,  285. 

Calorimeter,  combustion,  289. 
Capillary  electrometer,  458. 
Carbon  dioxide,  assimilation  of,  523. 
Catalysis,  379. 

mechanism  of,  352. 

negative,  382. 
Cataphoresis,  251. 

Cathode  particle,  charge  carried  by, 
38. 

ratio  of  charge  to  mass  of,  38. 

velocity  of,  36. 
Cathode  rays,  33. 

properties  of,  33. 
Cathodo-luminescence,  512. 
Cells,    electromotive    force    of    con- 
centration, 411. 

galvanic,  446. 

gas,  485. 

photo-electric,  525. 

reversible,  448. 

standard,  457. 

storage,  488. 

Chemical  constitution,  132, 142. 
Chemi-luminescence,  512. 
Colloidal  solutions,  238. 

density  of,  245. 

electrical  conductance  of,  253. 

osmotic  pressure  of,  247. 

preparation  of,  274. 

surface  tension  of,  247. 

viscosity  of,  246. 
Colloids,  237. 


533 


534 


INDEX  OF  SUBJECTS 


Colloids,  molecular  weight  of,  249. 

precipitation  by  electrolytes,  254. 

surface  energy  of,  271. 
Components,  338. 
Compounds,  2,  7. 

Compressibilities  of  solid  elements,  159. 
Concentration  and  activity,  478. 

elements,  465. 
Conductance,  and  ionization,  412. 

at  high  temperatures  and  pressures, 
417. 

determination  of,  400. 

equivalent,  398. 

molar,  398. 

specific,  398. 

Conductance    of    difficultly    soluble 
salts,  415. 

of  different  substances,  402. 

of  fused  salts,  420. 

of  non-aqueous  solutions,  418. 

temperature  coefficient  of,  416. 
Conduction    of    electricity    through 

gases,  31. 
Connection    between    gaseous    and 

liquid  states,  104. 
Consecutive  reactions,  376. 
Corresponding  conditions,  110. 
Co- volume,  121. 
Critical  pressure,  105. 

temperature,  105. 

volume,  105. 
Crookes'  dark  space,  32. 
Cryohydrate,  347. 

Crystal  form  and  chemical  composi- 
tion, 158. 

Crystallography,  154. 
Crystalloids,  237. 
Crystals,  liquid,  165. 

mixed,  14. 

properties  of,  156. 
Crystallization  methods,  275. 

Decomposition  potential,  495. 
Degree  of  dissociation,  calculation  of, 
91. 


Degree  of  dissociation,  freedom,  339. 

Density  of  colloidal  solutions,  245. 

Dialysis,  237. 

Dielectric  constants,  150. 

Diffusion,  coefficient  of,  206. 

Dilution  law,  422. 

Disintegration  hypothesis,  50. 

series,  52. 
Dispersity,  238. 
Dispersoids,  239,  245. 

classification  of,  244. 
Dissociation  constant,  320. 

hypothesis,  437. 

in  solution,  226. 
Distillation  fractional,  178. 

steam,  179. 
Distribution  of  a  solute  between  two 

immiscible  solvents,  335. 
Drop  weight,  149. 

Electrical    conductance    of    colloidal 
solutions,  253. 

double  layer,  452. 
Electrode,  dropping,  463. 

hydrogen,  464. 

normal,  461. 

null,  463. 
Electrodes,  of  first  type,  468. 

of  second  type,  468. 
Electrpendosmosis,  250. 
Electro-luminescence,  512. 
Electrolytes,    ionization    of    strong, 
429. 

mixtures  with  a  common  ion,  427. 
Electrolytic  dissociation  theory,  227, 
229. 

separation  of  metals,  502. 
Electrometer  capillary,  458. 
Electromotive  force,  measurement  of, 

455. 

Electrons,  sources  of,  41. 
Electron  theory,  31. 
Elements,  2. 

classification  of,  20. 

oxidation  and  reduction,  482. 


INDEX  OF  SUBJECTS 


535 


Elements,  periodic  series  of,  24. 

Elevation  of  boiling  point,  213. 

Emanations,  48. 

Emission  and  absorption,  506. 

Emulsions,  239. 

Emulsoids,  239. 

action  of  heat  on,  257. 

precipitation  of,  257. 
Energy  evolved  by  radium,  56. 

distribution  in  spectrum,  509. 
Equation  of  condition,  reduced,  111. 
Equation  of  Nernst-Lindemann,  164. 

van  der  Waals,  81,  107. 
Equilibrium  constant,  316. 

variation  with  temperature,  324. 
Equilibrium,  homogeneous,  312. 

in  homogeneous  gaseous  systems, 
317. 

in  liquid  systems,  322. 
Equivalent,  chemical,  6. 

electrochemical,  390. 
Etch  figures,  158. 
Eutectic,  353. 

Faraday  dark  space,  32. 
Ferments,  inorganic,  381. 
Fluorescence,  511. 
Formation,  heat  of,  294. 
Freezing  point,  lowering  of,  217. 
of  concentrated  solutions,  233. 
Fusion,  heat  of,  153. 

Galvanic  cells,  446. 
Gas,  ideal,  75. 

perfect,  75. 
Gases,  adsorption  of,  268. 

specific  heat  of,  100. 
Gas  constant,  molecular,  73. 

specific,  73. 

Gas  laws,  deviations  from,  74. 
Gaseous  and  liquid  states,  connection 

between,  104. 
Gaseous  molecule,  mean  velocity  of 

translation  of,  80. 
Gelation,  239. 


Gels,  elastic,  265. 

hydration  and  dehydration  of,  263. 

non-elastic,  264. 

physical  properties  of,  262. 
Gold-number,  259. 

Hsematocrit  method,  205. 
Heat,  of  combustion,  302. 

of  dilution,  297. 

of  dissociation  of  solids,  334. 

of  formation,  294. 

of  fusion,  153. 

ol  imbibition,  266. 

of  ionization,  397,  432. 

of  neutralization,  305. 

of  reaction,  variation  with  tempera- 
ture, 300. 

of  solution,  295. 

of  vaporization,  11$. 
Helium  atoms  and  a-particles  iden- 
tical, 55. 

Helium,  rate  of  production  of,  56. 
Helix  of  de  Chancourtois,  21. 
Heterogeneous  systems,  328. 
Homogeneous  gaseous  systems,  equi- 
librium in,  317. 
Hydrogel,  328. 
Hydrolysis,  437. 

determination  of,  442,  480. 
Hydrosol,  238. 

Imbibition,  266. 

in  solutions,  267. 

velocity  of,  266. 
Immiscibility,  178. 
Inorganic  ferments,  381. 
Ionic  product,  433. 
Ionization  constant,  422. 

of  water,  443. 
Ionization  of  gases,  45. 
Ionization,  heat  of,  432,  484. 

influence  of  substitution  on,  435. 

of  water,  487. 
lonogen,  229. 
Ions,  45. 


536 


INDEX  OF  SUBJECTS 


Ions,  absolute  velocity  of,  409. 

existence  of  free,  391 . 

migration  of,  393. 
Isohydric  solutions,  428. 
Isomorphism,  14. 
Isothermals,  105. 
Isotopes,  63. 
Isotonic  coefficient,  203. 

Kinetic  equation,  deductions  from,  78. 

derivation  of,  77. 

theory  of  gases,  100. 
Kinetics,  chemical,  359. 

Labile  solutions,  183. 
Law  of  Beer,  141,  515. 

of  Boyle,  73,  78. 

of  combining  .proportions,  5. 

of  conservation  of  mass,  3. 

of  constant  heat  summation,  290. 

of  definite  proportions,  4. 

of  Dulong  and  Petit,  11. 

of  Faraday,  389. 

of  Gay-Lussac,  73,  79. 

of  Graham,  80. 

of  Grotthuss,  513. 

of  Guldberg  and  Waage,  314. 

of  Henry,  170,  171. 

of  Hess,  290,  305. 

of  Jurin,  144. 

of  Kirchhoff,  506. 

of  Kohlrausch,  406. 

of  Lavoisier  and  Laplace,  290. 

of  mass  action,  314. 

of  Mitscherlich,  14,  15. 

of  molecular  displacement,  283. 

of  multiple  proportions,  5. 

of  Neumann,  14. 

of  octaves,  22. 

of  Ohm,  388. 

of  Raoult,  209. 

of  Richter,  4. 

of  Stefan-Boltzmann,  507. 

of  volumes,  8. 

of  Wien,  507. 


Light,  action  on  silver  halides,  521. 
Liquefaction  of  gases,  112. 
Liquids,  characteristics  of,  104. 

refractive  power  of,  123. 

vapor  pressure  of,  117. 
Liquid  systems,  equilibrium  in,  322. 
Luminescence,  511. 
Lyotrope  series,  239. 

Magnetic  rotation,  138. 
Mass,  conservation  of,  3. 
Mass  action,  law  of,  314. 
Maximum  work,  principle  of,  308. 
Mechanism  of  catalysis,  382. 
Membranes,  semi-permeable,  191. 
Metastable  solutions,  183. 
Microns,  238. 
Miscibility,  complete,  174. 

partial,  172. 

Mixtures,  specific  refraction  of,  128. 
Molar  volume,  10. 

Molecular  gas   constant,   evaluation 
of,  74. 

magnetic  rotation,  139. 

refraction,  126. 

rotation,  131. 

vibration,  142. 

volume,  120. 

weight,  83,  146. 

weight  by  boiling  point  method, 
215. 

weight  by  freezing  point  method, 
220. 

weight  in  solution,  222. 

weight  of  colloids,  249. 

weight  in  solution,  222. 

vibration,  142. 

volume,  120. 

Neutralization,  heat  of,  305. 
Nitrogen    peroxide,    dissociation    of, 

93. 
Non-dissociating  solvent,  solution  of 

solid  in,  336. 
Normal  electrodes,  461. 


INDEX  OF  SUBJECTS 


537 


Octaves,  law  of,  22. 
Optical  activity,  132. 
Osmotic  pressure,  187,  191. 

and  boiling  point  elevation,  216. 

and  diffusion,  206. 

and  freezing  point  depression,  221. 
Osmotic    pressure    and    lowering    of 
vapor  pressure,  211. 

comparative  values  of,  201. 

measurement  of,  189. 

of  colloidal  solutions,  247. 

recent    work    on    direct    measure- 
ment of,  194. 

theoretical  value  of,  192. 
Oxidation    and    reduction    elements, 
482. 

Particles,  a-,  55. 
Peptization,  276. 
Periodic  law,  22,  64. 

applications  of,  26. 

defects  in,  29. 
Periodicity  of  physical  properties,  25. 

of  radio-elements,  61. 
Phase  rule,  338. 

derivation  of,  339. 
Phosphorescence,  511. 
Photochemical  action,  513. 

after  effect,  523. 

extinction,  515. 

induction,  518. 

reaction,  kinetics  of,  516. 

sensitization,  522. 

synthesis,  524. 
Photo-electric  cells,  525. 
Photo-stationary  state,  518. 
Physical  properties  of  ionized  solu- 
tions, 231. 

Plasmolytic  methods,  201. 
Polarization,  492. 

capacity  of  electrodes,  494. 

electromotive  force  of,  492. 

theory  of,  497. 

Polarized  light,  rotation  of  plane  of, 
129. 


Polymorphism,  158. 
Potential,  between  metal  and  solu- 
tion, 464. 

decomposition,  495. 

difference  at  junction  of  two  solu- 
tions, 469. 

difference  at  liquid  junctions,  472. 

normal  electrode,  475. 
Precipitation  and  valence,  256. 
Precipitation  of  colloids  by  electro- 
lytes, 254. 

of  emulsoids,  257. 

of  suspensoids,  254. 
Preparation    of    colloidal    solutions, 

274. 
Principle  of  maximum  work,  308. 

of  Soret,  208. 
Properties  of  cathode  rays,  33. 

of  gels,  262. 
Proportions,  law  of  combining,  5. 

law  of  definite,  4. 

law  of  multiple,  5. 
Protective  colloids,  258. 
Prout's  hypothesis,  20. 

Quantum  Theory,  164. 

Radiant  energy,  504. 

source  of,  505. 
Radiations,  nature  of,  46. 

phosphorescence  induced  by,  46. 

photographic  action  of,  45. 
Radioactive  constant,  50. 

equilibrium,  52. 

substances,  44. 
Radioactivity,  42. 

discovery  of,  42. 

Radio-elements,  periodicity  of,  61. 
Radio-luminescence,  512. 
Radium,  43. 

discovery  of,  43. 

energy  evolved  by,  56. 
Rays,  a-,  /3-,  and  7-,  47. 
Ratio  of  charge  to  mass,  38. 

of  specific  heats,  96. 


538 


INDEX  OF  SUBJECTS 


Ratio  of  charge  to  mass,  of  specific 

heats,  determination  of,  97,  98. 
Reaction,  determination  of  order  of, 
374. 

influence  of  solvent  on  velocity  of, 
378. 

velocity,  359,  374,  377. 
Reactions,  at  constant  pressure,  298. 

at  constant  volume,  298. 

bimolecular,  366. 

counter,  374. 

heterogeneous,  velocity  of,  376. 

of  first  order,  364. 

of  higher  orders,  373. 

of  second  order,  366. 

of  third  order,  370. 

photochemical,    classification     of, 
519. 

side,  374. 

trimolecular,  370. 
Reciprocal  precipitation,  260. 
Refraction,  index  of,  124. 

molecular,  126. 

specific,  126. 
Residual  current,  495. 
Resistance  capacity,  401. 
Reversible  cells,  448. 
Rotation,  magnetic,  139. 

molecular,  131. 

of  plane  polarized  light,  129. 

specific,  131. 

Salt  solutions,  thermoneutrality  of, 

305. 

Saturated  solution,  181. 
Semi-permeable  membranes,  187. 
Side  reactions,  374. 
Solation,  239. 
Solids,  general  properties  of,  153. 

heat  capacity  of,  161. 
Solubility  coefficient,  171. 
i       product,  433. 
Solutes,  abnormal,  225. 
Solution,  association  in,  225. 

dissociation  in,  226. 


Solution,  heat  of,  295. 

methods,  276. 

pressure,  450. 
Solutions,  167. 

absorption  in,  269. 

classification  of,  167. 

colloidal,  238. 

conductance  of  non-aqueous,  418. 

imbibition  in,  267. 

of  gases  in  gases,  167. 

of  gases  in  liquids,  167. 

of  liquids  in  liquids,  172. 

of  solids  in  liquids,  181. 

isohydric,  428. 

labile,  183. 

metastable,  183. 

properties  of  ionized,  231. 

saturated,  169. 

solid,  184. 

supersaturated,  169. 

volume-normal,  196. 

weight-normal,  196. 
Solvate  theory,  234. 
Solvent,  ionizing  power  of,  419. 
Specific  heat,  92,  93. 

and  atomic  weight,  11. 

at  constant  pressure  and  volume, 
93. 

of  gases,  100. 

of  solid  elements,  11,  161. 

ratio  of,  96. 
Specific  refraction,  126. 

rotation,  131. 
Spectra,  absorption,  139. 
Standard  cells,  457. 
Steam  distillation,  179. 
Strength  of  acids  and  bases,  424. 
Sublimation,  153. 
Submicrons,  239. 
Sulphur,    equilibrium    between    the 

phases  of,  343. 
Surface  concentration,  272. 
Surface  energy  of  colloids,  271. 
Surface  tension,  144,  146,  149. 

of  colloidal  solutions,  247. 


INDEX  OF  SUBJECTS 


539 


Suspensions,  239. 
Suspensoids,  239. 

precipitation  of,  254. 
Systems,  three-component,  356. 

two-component,  345. 

Theorem  of  Le  Chatelier,  309. 
Theory,  electron,  31. 

kinetic,  100. 

of    electrolytic    dissociation,    227, 
229. 

of  solvation,  234. 
Thermal  units,  286. 
Thermochemical  equations,  287. 

measurements,  288. 
Thermo-luminescence,  512. 
Thermoneutrality  of  salt   solutions, 

304. 

Transition  point,  340. 
Transmission  coefficient,  514. 
Transport  numbers,  395. 

determination  of,  395. 

electrometric  determination  of,  477. 
Triads,  of  Dobereiner,  21. 
Tribo-luminescence,  512. 
Tri molecular  reactions,  370. 
Triple  point,  342. 
Tyndall  phenomenon,  241. 

Ultrafiltration,  242. 
Ultramicroscope,  241. 
Unimolecular  reactions,  361,  364. 
Units,  electrical,  388. 

Valence,  15. 
electrometric  determination  of,  476. 


Vapor  density,  83. 

abnormal,  88. 

determinations  of,  86. 
Vaporization,  heat  of,  118. 
Vapor  pressure,  117. 

and  osmotic  pressure,   connection 
between,  211. 

lowering  of,  209. 

of  liquids,  117. 

Variation    of    equilibrium    constant 
with  temperature,  337. 

of  heat  of  reaction  with  tempera- 
ture, 300. 
Velocity  constant,  315. 

of  gaseous  molecule,  80. 

of  imbibition,  266. 
Vibration,  curves  of  molecular,  142. 
Viscosity  of  colloidal  solutions,  246. 
Voltaic  pile,  385. 
Volume,  molar,  10. 
Volumes,  Gay-Lussac's  law  of,  8. 

Water,  dissociation  of,  414. 

equilibrium  between  phases  of, 
340. 

ionization  constant  of,  443. 

primary  decomposition  of,  501. 
Weight,  atomic,  8,  10,  17,  18. 

combining,  5,  8. 

gram-molecular,  10,  17. 

molar,  10,  17. 

X-rays  and  atomic  structure,  58. 
and  crystal  structure,  159. 
spectra,  60. 


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